<?xml version='1.0' encoding='UTF-8'?><rss xmlns:atom="http://www.w3.org/2005/Atom" xmlns:openSearch="http://a9.com/-/spec/opensearchrss/1.0/" xmlns:blogger="http://schemas.google.com/blogger/2008" xmlns:georss="http://www.georss.org/georss" xmlns:gd="http://schemas.google.com/g/2005" xmlns:thr="http://purl.org/syndication/thread/1.0" version="2.0"><channel><atom:id>tag:blogger.com,1999:blog-6615370946101387199</atom:id><lastBuildDate>Wed, 28 Aug 2024 06:58:00 +0000</lastBuildDate><category>K map</category><category>5G</category><category>ASK
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OFDMA</category><category>Types of modulation</category><category>Types of printers</category><category>Types of processors</category><category>Types of storage devices</category><category>Up and Down counters</category><category>advantages of digital electronics</category><category>history of computers</category><category>java</category><category>op amp</category><title>knowitanything</title><description>Engineering and Technology</description><link>https://www.knowitanything.com/</link><managingEditor>noreply@blogger.com (study engineering)</managingEditor><generator>Blogger</generator><openSearch:totalResults>78</openSearch:totalResults><openSearch:startIndex>1</openSearch:startIndex><openSearch:itemsPerPage>25</openSearch:itemsPerPage><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-1284176590926880605</guid><pubDate>Mon, 21 Aug 2023 08:13:00 +0000</pubDate><atom:updated>2023-08-21T13:43:59.209+05:30</atom:updated><title>Signal generator and Wave shaping</title><description><p>Let's look at the signal generator and wave shaping concepts:</p><p>Signal generator: A signal generator is a sort of electronic equipment that generates various types of electrical signals. These signals can be utilized in electronics and communications for testing, troubleshooting, calibration, and experimentation. To imitate real-world signals, signal generators can generate a variety of waveforms, frequencies, and amplitudes.</p><p>Types of Signals Produced:</p><p></p><ol style="text-align: left;"><li>Sine Wave: A smooth waveform with positive and negative values that oscillates. It is frequently used to test audio equipment and determine frequency response.</li><li>Square Wave: A square wave is a waveform that abruptly shifts between two voltage levels (high and low). It can be used to test digital circuitry and clock signals.</li><li>Triangle Wave: A waveform with a linear ramp up and down. It is used for waveform generation and modulation.</li><li>Sawtooth Wave: A waveform that increases linearly and then abruptly resets. It is utilized in modulation and synchronization.</li><li>Pulse Wave: A pulse wave is a waveform with controlled width and period that consists of high and low levels. It's commonly seen in digital systems and timing applications.</li></ol><p></p><p>Wave Shaping is the process of altering the shape of a waveform to get the desired output. This can be accomplished by the use of electronic circuits that alter the amplitude, frequency, phase, or shape of the original waveform.</p><p>Wave Shaping Examples:</p><p></p><ol style="text-align: left;"><li>Amplitude Modulation (AM): AM is the process of multiplying a carrier signal (often a sine wave) by a modulating signal (typically an audio waveform). The result is a changing amplitude waveform that is employed in radio transmission.</li><li>Frequency Modulation (FM): FM alters the frequency of a carrier signal based on a modulating signal rather than modifying its amplitude. It's regularly heard on the radio.</li><li>Phase Shifts Oscillators: These circuits use phase shifts in a feedback loop to generate oscillations. They are used to create sine waveforms.</li><li>Clipping and clamping circuits: They alter the shape of a waveform by deleting or limiting specific components. Clipping circuits eliminate portions of the waveform that are above or below a specific threshold, whereas clamping circuits shift the waveform vertically.</li><li>Waveform Synthesis: Using addition, multiplication, or modulation to combine several waveforms, you can construct complex and distinctive wave shapes for musical synthesis and sound creation.</li></ol><p></p><p>Wave shaping techniques are critical for transforming raw signals into formats appropriate for transmission, processing, and other uses. Signal generators and wave shaping circuits are essential tools in signal processing and electronics.</p><p><br /></p></description><link>https://www.knowitanything.com/2023/08/signal-generator-and-wave-shaping.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7576735019102376777</guid><pubDate>Mon, 21 Aug 2023 07:51:00 +0000</pubDate><atom:updated>2023-08-21T13:21:04.432+05:30</atom:updated><title>Half wave and Full wave rectifier</title><description><p>A half-wave rectifier permits only one half of an alternating current waveform to flow through while blocking the other. By converting the alternating current (AC) signal to a unidirectional (DC) signal, a full-wave rectifier allows both halves to pass through. Do you want to learn more about how these rectifiers function or their applications?</p><p>More information about half-wave and full-wave rectifiers is here:</p><p>Half-Wave Rectifier:&nbsp;</p><p></p><ul style="text-align: left;"><li>A half-wave rectifier permits only one half of an alternating current waveform to flow through while blocking the other half.</li><li>It is made up of a diode that is linked in series with the AC input source. When the input voltage is positive (or negative), the diode only conducts, thereby cutting off the negative (or positive) component of the AC signal.</li><li>Because it only delivers output during half of the input cycle, the output of a half-wave rectifier is not as smooth as that of a full-wave rectifier.</li><li>Half-wave rectifiers are commonly employed in low-cost applications such as battery chargers and tiny power supplies due to their simplicity.</li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu7iDuOfTt8X7MqZGmhcEBq3TwJX54YOpWhFLRd2LkbQnLCgYaEE_9knNnGUccIHKEky2ZogHCaOVponoEXaZEpZJGJpBbg-Kl4hZaJ0sU8XiCEo-x2dWr6q58GL64HIwnJw44ykTOLM8dQRrKX2esXJ-BaAN9HbL5Yl7TTHA-7rRObuc6hm0aTQ2F4K4/s640/images%20(24).jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="268" data-original-width="640" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu7iDuOfTt8X7MqZGmhcEBq3TwJX54YOpWhFLRd2LkbQnLCgYaEE_9knNnGUccIHKEky2ZogHCaOVponoEXaZEpZJGJpBbg-Kl4hZaJ0sU8XiCEo-x2dWr6q58GL64HIwnJw44ykTOLM8dQRrKX2esXJ-BaAN9HbL5Yl7TTHA-7rRObuc6hm0aTQ2F4K4/s320/images%20(24).jpeg" width="320" /></a></div><div><br /></div><div><br /></div><div>Full-Wave Rectifier:&nbsp;</div><div><ul style="text-align: left;"><li>A full-wave rectifier converts both the positive and negative half of an alternating current waveform into direct current voltage.</li><li>Full-wave rectifiers are classified into two types: bridge rectifiers and center-tapped rectifiers.</li><ul><li>Bridge Rectifier: A bridge rectifier is made up of four diodes placed in a bridge arrangement that efficiently transforms both halves of an alternating current input to direct current.</li><li>Center-Tapped Rectifier: Splits the input waveform into two halves using a center-tapped transformer. Each half is rectified by two diodes, yielding a full-wave rectified output.</li></ul></ul></div><div><ul style="text-align: left;"><li>A full-wave rectifier produces DC output during both halves of the input cycle, resulting in smoother output than a half-wave rectifier.</li><li>Full-wave rectifiers are widely utilized in power supply applications such as electronic devices, industrial equipment, and so on.</li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibCZ34Mz3T4p5o7MEZa6dy_ttNRoBZkms0VJtsLkFm_cxWLkSPIpYishLW3YivgN2MYpRh6AqEeCjqMB67R5FLhgIQuUdwPJxUWFbH07Owfqh8bPLKylTYJvnHIEytB3Xcxio5JTQGE2STv6skIyEYRNqYfmMKNM5qGrBFUHJAUDSXJXLcD4kuXrr7Gps/s600/images%20(25).jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="350" data-original-width="600" height="187" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibCZ34Mz3T4p5o7MEZa6dy_ttNRoBZkms0VJtsLkFm_cxWLkSPIpYishLW3YivgN2MYpRh6AqEeCjqMB67R5FLhgIQuUdwPJxUWFbH07Owfqh8bPLKylTYJvnHIEytB3Xcxio5JTQGE2STv6skIyEYRNqYfmMKNM5qGrBFUHJAUDSXJXLcD4kuXrr7Gps/s320/images%20(25).jpeg" width="320" /></a></div><div><br /></div><div><br /></div><div>Half-wave and full-wave rectifiers are both essential components in converting AC voltage to DC voltage and allowing devices to function with a reliable DC power source. Their decision is influenced by considerations such as efficiency, cost, and application needs.</div></div><p></p><p><br /></p></description><link>https://www.knowitanything.com/2023/08/half-wave-and-full-wave-rectifier.html</link><author>noreply@blogger.com (study engineering)</author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu7iDuOfTt8X7MqZGmhcEBq3TwJX54YOpWhFLRd2LkbQnLCgYaEE_9knNnGUccIHKEky2ZogHCaOVponoEXaZEpZJGJpBbg-Kl4hZaJ0sU8XiCEo-x2dWr6q58GL64HIwnJw44ykTOLM8dQRrKX2esXJ-BaAN9HbL5Yl7TTHA-7rRObuc6hm0aTQ2F4K4/s72-c/images%20(24).jpeg" height="72" width="72"/><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7399254121362151387</guid><pubDate>Wed, 16 Aug 2023 09:38:00 +0000</pubDate><atom:updated>2023-08-21T13:24:18.250+05:30</atom:updated><title>8086 Microprocessor Architecture</title><description><p><span style="font-size: medium;">A 16-bit microprocessor called the Intel 8086 was first released by Intel in 1978. It was the founding member of the x86 family of processors and had a significant impact on the development of microprocessors. The 8086's architecture served as the model for other CPU generations that followed.</span></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVwuB0UtA5vfk3q8XGZE2haxmsSmNAuvc-AKlu5wFY9SMUx-ZnKf2yidCJabUg_yFhXr847OsTnbl5hrb89GRy8YgscElnos08Xyl7K4TkH_mfIukx5O0Ii6KarbvxGvTVYWnMPhfNXe6eyMTtuyiRjsXCCJtIfv4u0KZlhgy-JudmNlZBg7tewwcjmus/s600/images%20(27).jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="422" data-original-width="600" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVwuB0UtA5vfk3q8XGZE2haxmsSmNAuvc-AKlu5wFY9SMUx-ZnKf2yidCJabUg_yFhXr847OsTnbl5hrb89GRy8YgscElnos08Xyl7K4TkH_mfIukx5O0Ii6KarbvxGvTVYWnMPhfNXe6eyMTtuyiRjsXCCJtIfv4u0KZlhgy-JudmNlZBg7tewwcjmus/s320/images%20(27).jpeg" width="320" /></a></div><br /><span style="font-size: medium;"><br /></span><p></p><p><span style="font-size: medium;">The 8086 microprocessor architecture's main characteristics and components are listed below:</span></p><p><span style="font-size: medium;">1. 16-Bit Architecture: The 8086 uses 16-bit data and addresses since it is a 16-bit microprocessor. Compared to its predecessors, this made it able to process more data.</span></p><p><span style="font-size: medium;">2. Four general-purpose registers (AX, BX, CX, and DX), a stack pointer (SP), a base pointer (BP), a source index (SI), and a destination index (DI) are among the 16-bit registers available on the 8086. These registers can be used for storing data, doing calculations, and addressing, among other things.</span></p><p><span style="font-size: medium;">3. Segmentation: The 8086 architecture uses segmentation, which is one of its distinguishing characteristics. Segment registers (CS, DS, SS, and ES) are used to address each of the 64KB segments that make up memory. A physical memory address is made up of a segment register and an offset address. The 8086 was able to address memory that was larger than 64KB because to this segmentation strategy.</span></p><p><span style="font-size: medium;">4. Addressing Modes: The 8086 offers immediate, register, direct, and indirect addressing, among other addressing modes. It is possible to change data and addresses in a variety of ways because to this flexibility.</span></p><p><span style="font-size: medium;">5. Instruction Set: The 8086 contains a comprehensive instruction set that includes many different operations for manipulating data, performing arithmetic and logic operations, branching, and more. Instructions for data transfer, arithmetic, logic, control transfer, and string manipulation are some examples of common categories for instructions.</span></p><p><span style="font-size: medium;">6. Memory: The 8086 is capable of addressing memory up to 1MB. With a 20-bit address bus, it can address 220 (1,048,576) different memory locations. The full 1MB memory region can be addressed, however it can be difficult because of the segmentation.</span></p><p><span style="font-size: medium;">7. Pipelining: The 8086 has a basic version of pipelining where the following instruction is executed before the previous one is finished. This aids in raising the processor's total throughput.</span></p><p><span style="font-size: medium;">8. Clock Speed: The initial 8086 processor ran at a 5 MHz clock rate. As time went on, Intel introduced a number of revisions with faster clock rates, and finally, more sophisticated generations of processors were created.</span></p><p><span style="font-size: medium;">9. Compatibility: The x86 family of processors, which is still extensively used today, was founded on the 8086 and its descendants. On current x86 processors, 8086 software can run to varied degrees of compatibility.</span></p><p><span style="font-size: medium;">The x86 architecture that came after the 8086 architecture was greatly influenced by the 8086 architecture and had a huge impact on the computing industry.</span></p><p><span style="font-size: medium;">Let's examine some further features of the 8086 microprocessor architecture in more detail:</span></p><p><span style="font-size: medium;">1. Real Mode and Protected Mode: The 8086 was designed to function in "real mode," which was an imitation of the actions of prior 8-bit processors. Later versions of the architecture added "protected mode," which provided improved virtual memory, multitasking, and memory management features. Modern operating systems can use memory and resources more effectively thanks to protected mode.</span></p><p><span style="font-size: medium;"><span style="background-color: white;"><span color="var(--tw-prose-bold)" face="Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; white-space-collapse: preserve;">2. Interrupts and Interrupt Vector Table:</span><span face="Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;" style="color: #374151; white-space-collapse: preserve;"> </span></span>The 8086 supports a variety of hardware interrupts and software interrupts. The processor temporarily interrupts its ongoing work in response to an interrupt, and then jumps to a specific region of memory known as the Interrupt Vector Table (IVT). The addresses of the interrupt service routines (ISRs) that manage the interrupts are listed in this table.</span></p><p><span style="font-size: medium;">3. BIU and EU: The Bus Interface Unit (BIU) and the Execution Unit (EU) are the two primary conceptual components of the 8086 architecture. While the EU handles the actual execution of instructions, the BIU is in charge of retrieving data and instructions from memory. The processor's ability to multitask and use pipelining is enhanced by this division.</span></p><p><span style="font-size: medium;">4. Assembler Directives: The assembly language used by the 8086 must be understood in order to program it. Using assembler directives, you can provide the assembler more details that will assist it manage memory allocation, segment definitions, and data formatting. Due to the architecture's segmented memory concept, this is crucial.</span></p><p><span style="font-size: medium;">5. String Instructions: The 8086 provided specialized instructions for manipulating strings that work with arrays of bytes or words. It is more effective to work with data blocks when using these instructions, which include MOVS, CMPS, and SCAS. They are especially helpful for copying, comparing, and searching strings.</span></p><p><span style="font-size: medium;">6. Coprocessors Support: The Intel 8087 math coprocessor, which could carry out floating-point arithmetic operations, was supported by the 8086 architecture. This significantly increased the efficiency of numerical computations in high precision applications.</span></p><p><span style="font-size: medium;">7. Instructions Set Extension: As the 8086 developed, Intel added new instructions and improved existing ones in later generations such the 80186 and 80286. These upgrades brought about improved memory management, enhanced capabilities, and more powerful computing.</span></p><p><span style="font-size: medium;">8. Compatibility Modes: In order to enable older software developed for the original 8086 and its successors to work on modern processors, newer x86 family CPUs added modes like "Virtual 8086 Mode" and "Compatibility Mode."</span></p><p><span style="font-size: medium;">9. Legacy Effect: The 8086's architecture has endured despite its introduction many years ago. One of the most popular processor architectures for desktops, laptops, servers, and embedded systems continues to be the x86 family of processors, which is descended from the 8086.</span></p><p><span style="font-size: medium;">10. Instruction Encoding: Variable-length instruction encoding is used by the 8086. This indicates that various instructions have different lengths, which makes the decoding process more difficult than with instruction sets with fixed lengths. But this method enables a condensed depiction of instructions.</span></p><p><span style="font-size: medium;">11. Reverse Compatibility: Despite being far more powerful and sophisticated than the original 8086, newer x86 processors nonetheless maintain some degree of backward compatibility with it. As a result, software created for early systems can still be used.</span></p><p><span style="font-size: medium;">The architecture of the Intel 8086 paved the way for the creation of contemporary computing systems and was significant in influencing the advancement of microprocessors and computer technology.</span></p></description><link>https://www.knowitanything.com/2023/08/8086-microprocessor-architecture.html</link><author>noreply@blogger.com (study engineering)</author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVwuB0UtA5vfk3q8XGZE2haxmsSmNAuvc-AKlu5wFY9SMUx-ZnKf2yidCJabUg_yFhXr847OsTnbl5hrb89GRy8YgscElnos08Xyl7K4TkH_mfIukx5O0Ii6KarbvxGvTVYWnMPhfNXe6eyMTtuyiRjsXCCJtIfv4u0KZlhgy-JudmNlZBg7tewwcjmus/s72-c/images%20(27).jpeg" height="72" width="72"/><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7603104360461834738</guid><pubDate>Wed, 16 Aug 2023 09:02:00 +0000</pubDate><atom:updated>2023-08-21T13:23:33.526+05:30</atom:updated><title>8085 Microprocessor Architecture</title><description><p><span style="font-size: medium;">An 8-bit microprocessor called the 8085 was first released by Intel in 1976. It is a member of the 8080/8085 microprocessor family and is frequently used in many different applications, including early personal computers and embedded devices. Key elements of the microprocessor architecture for the 8085 include:</span></p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPu4nVwXnka5C1GwsJmlxL_PD21z0ThTou49lOsNRF5A14mWaNDtJ_dmaTj4Km2MCD2s3Zc5p3_Xhcq4xobl5LvZ4odIgAuLeGFBTwxHDr--R_dZMn8xPMHoR-R1MLBpA5G6lleh6kQL8ZXoXTfmCybox2M_HO31Jte_kc_g9hX0zgEYVuXRPHQC4c6lQ/s600/images%20(26).jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="426" data-original-width="600" height="227" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPu4nVwXnka5C1GwsJmlxL_PD21z0ThTou49lOsNRF5A14mWaNDtJ_dmaTj4Km2MCD2s3Zc5p3_Xhcq4xobl5LvZ4odIgAuLeGFBTwxHDr--R_dZMn8xPMHoR-R1MLBpA5G6lleh6kQL8ZXoXTfmCybox2M_HO31Jte_kc_g9hX0zgEYVuXRPHQC4c6lQ/s320/images%20(26).jpeg" width="320" /></a></div><br /><span style="font-size: medium;"><br /></span><p></p><p><span style="font-size: medium;">1. Accumulator: The main working register for arithmetic and logical operations is the accumulator, an 8-bit register. During processing, it temporarily stores the data.</span></p><p><span style="font-size: medium;">2. General Purpose Registers: There are six 8-bit general-purpose registers with the labels B, C, D, E, H, and L. These registers can be used for a variety of tasks, such as data storage and manipulation.</span></p><p><span style="font-size: medium;">3. Stack Pointer: The stack pointer (SP) is a 16-bit register that points to the top of the memory stack. During calls to subroutines, the stack is utilized to store return addresses and data.</span></p><p><span style="font-size: medium;">4. Program Counter (PC): The location of the following instruction to be fetched and performed is stored in the program counter, a 16-bit register.</span></p><p><span style="font-size: medium;">5. Flag Register: The results of arithmetic and logical operations are used to set or clear various status flags in the flag register. Sign (S), Zero (Z), Auxiliary Carry (AC), Parity (P), and Carry (CY) flags are among them.</span></p><p><span style="font-size: medium;">6. Instruction Register: The instruction currently being executed is stored in the instruction register (IR).</span></p><p><span style="font-size: medium;">7. Control and Status Register: These signals regulate a number of microprocessor functions, including read and write operations, interrupt management, and clock synchronization.</span></p><p><span style="font-size: medium;">8. ALU (Arithmetic Logic Unit): The ALU operates on data stored in the accumulator and other registers using logic and arithmetic operations. Addition, subtraction, logical AND/OR/XOR, and other operations are supported.</span></p><p><span style="font-size: medium;">9. Instruction Decoder: Decoding the fetched instruction and producing the required control signals for the instruction's execution are the responsibilities of the&nbsp;instruction decoder.</span></p><p><span style="font-size: medium;">10. Interrupts: Five interrupt lines are supported by the 8085 microprocessor: INTR, RST 7.5, RST 6.5, RST 5.5, and TRAP. Through the use of particular interrupt service routines, interrupts enable the microprocessor to react to outside events.</span></p><p><span style="font-size: medium;">A wide range of instructions for data movement, arithmetic and logical operations, branching, and I/O operations are included in the instruction set of the 8085 microprocessor. Binary-encoded instructions are sequentially read from memory, decoded, and carried out one at a time.</span></p><p><span style="font-size: medium;">Please be aware that although though the 8085 microprocessor was revolutionary at the time, it is now considered to be extremely old-fashioned. In terms of performance, architecture, and capabilities, modern microprocessors have undergone substantial development.</span></p><p><span style="font-size: medium;">Here are some more specifics regarding the architecture of the 8085 microprocessor:</span></p><p><span style="font-size: medium;">1. Data Bus and Address Bus: The 8085 microprocessor can access up to 64 KB of memory (2 x 16 = 65536 addresses) using a 16-bit address bus. For transporting data between the microprocessor and external memory or I/O devices, it contains an 8-bit data bus.</span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;">2. Instruction Set: Data transfer instructions (MOV), arithmetic instructions (ADD, SUB, etc.), logical instructions (AND, OR, XOR), branch instructions (JMP, CALL, RET), and input/output instructions (IN, OUT) are all included in the 8085 instruction set.</span></p><p><span style="font-size: medium;">3. Addressing Modes: Numerous addressing modes are supported by the 8085, which affect how operands are fetched for instructions. These include the addressing modalities of immediate, direct, indirect, register, and implied.</span></p><p><span style="font-size: medium;">4. Timing and Clock: A single-phase clock with a minimum frequency of 2.5 MHz is used by the CPU to run its operations. The length of time it takes for instructions to execute varies, and timing diagrams show how various signals are synced with the clock.</span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;">5. Serial Input/Output (SIO): The 8085 microprocessor has two serial input/output lines that can be utilized for serial data transmission protocols when interacting with external devices.</span></p><p><span style="font-size: medium;">6. Parallel Input/Output (PIO): The 8085 includes parallel input/output ports as well, which can be used to connect to external devices and transport data.</span></p><p><span style="font-size: medium;">7. Minimum Mode and Maximum Mode: There are two operating modes for the 8085 microprocessor: minimum mode and maximal mode. It functions as a solo CPU while in minimal mode. It is capable of functioning in multiprocessor mode.</span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;">8. Instruction Pipelining: Since the 8085 lacks instruction pipelining, every instruction must be retrieved, decoded, and executed individually. Modern processors, in contrast, use instruction pipelines to combine the fetch, decode, and execute stages for increased efficiency.</span></p><p><span style="font-size: medium;">9. Emulator and Simulators: There are several emulators and simulators that you may use to run 8085 programs on contemporary systems for learning and development reasons.</span></p><p><span style="font-size: medium;">10. Legacy: The architecture and instruction set of the 8085, though no longer widely used, served as the basis for later microprocessors and microcontrollers. It was essential in influencing how computing technology evolved.</span></p><p><span style="font-size: medium;"></span></p><p><span style="font-size: medium;">11. Books and Resources: You may learn a lot about the 8085 microprocessor's architecture, programming, and interface from a variety of books and online resources.</span></p><p><span style="font-size: medium;">It's crucial to keep in mind that the 8085 microprocessor is a vintage piece of hardware, and that while understanding its architecture is useful for historical context and educational purposes, current CPUs have made considerable strides in terms of performance, functionality, and architecture.</span></p></description><link>https://www.knowitanything.com/2023/08/8085-microprocessor-architecture.html</link><author>noreply@blogger.com (study engineering)</author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPu4nVwXnka5C1GwsJmlxL_PD21z0ThTou49lOsNRF5A14mWaNDtJ_dmaTj4Km2MCD2s3Zc5p3_Xhcq4xobl5LvZ4odIgAuLeGFBTwxHDr--R_dZMn8xPMHoR-R1MLBpA5G6lleh6kQL8ZXoXTfmCybox2M_HO31Jte_kc_g9hX0zgEYVuXRPHQC4c6lQ/s72-c/images%20(26).jpeg" height="72" width="72"/><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-2107103492533308746</guid><pubDate>Wed, 16 Aug 2023 08:46:00 +0000</pubDate><atom:updated>2023-08-16T14:16:03.307+05:30</atom:updated><title>Microprocessor Architecture</title><description><p><span style="font-size: medium;">The central processing unit (CPU) of a computer or other digital devices, a microprocessor, is referred to as having a microprocessor architecture. It includes a number of parts that cooperate to carry out computations and execute instructions, including registers, arithmetic logic units (ALUs), control units, cache memory, and more.</span></p><p><span style="font-size: medium;">Here are some essential ideas and elements that are frequently present in microprocessor architectures:</span></p><p><span style="font-size: medium;">Registers: The CPU uses these tiny, quick storage areas to temporarily store data while processing it. In holding operands, intermediate outcomes, and memory addresses, they are essential.</span></p><p><span style="font-size: medium;">Arithmetic Logic Unit (ALU): The addition, subtraction, multiplication, and division operations as well as logical (AND, OR, NOT) operations on data stored in registers are carried out by the arithmetic logic unit (ALU).</span></p><p><span style="font-size: medium;">Control Unit: The control unit oversees the execution of instructions by coordinating the data flow between various CPU and memory components. To govern the order of activities and the flow of data, it generates control signals.</span></p><p><span style="font-size: medium;">Memory Hierarchy: To speed up data access, modern microprocessors may have many levels of cache memory. These caches reduce the need to access slower main memory by storing frequently used data closer to the CPU.</span></p><p><span style="font-size: medium;"><span style="background-color: white; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; white-space-collapse: preserve;">Instruction Set Architecture (ISA):</span>A microprocessor can carry out a set of instructions known as the Instruction Set Architecture (ISA). The ISA specifies the activities the CPU is capable of carrying out as well as the register and memory addressing schemes that are employed.</span></p><p><span style="font-size: medium;">Pipelining: Pipelining is a technique that overlaps the execution of several instructions, enabling the CPU to begin processing a new instruction before the preceding one has finished. Performance as a whole and instruction throughput both increase as a result.</span></p><div><div><span style="font-size: medium;">Superscalar architecture: This design makes use of instruction-level parallelism by having many execution units on the microprocessor that may process numerous instructions at once.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Out-of-sequence Execution: This technique enables the CPU to change the sequence of instructions even though the original order is different in order to maximize execution by better utilizing the available execution units.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Branch Prediction: To forecast the results of conditional branches in the code, microprocessors frequently employ branch prediction algorithms. By doing this, pipeline stalls brought on by branching instructions are decreased.</span></div></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">RISC and CISC Architectures:&nbsp;RISC and CISC architectures are two different strategies for designing microprocessors. In order to achieve great performance, RISC (Reduced Instruction Set Computer) designs concentrate on using a small number of straightforward instructions. Complex instructions that can perform several operations in a single instruction are part of CISC (Complex Instruction Set Computer) architectures.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Von Neumann vs. Harvard Architectures: The Harvard design uses separate memory areas for instructions and data, enabling simultaneous access, as opposed to the Von Neumann architecture, which uses a single memory space for both instructions and data.</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">The architectures of microprocessors have advanced over time, becoming more complex and effective with each generation. The capabilities and performance of computers and other digital devices are fundamentally influenced by these architectures.</span></div></description><link>https://www.knowitanything.com/2023/08/microprocessor-architecture.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-5408955926619551074</guid><pubDate>Wed, 16 Aug 2023 08:31:00 +0000</pubDate><atom:updated>2023-08-16T14:01:39.542+05:30</atom:updated><title>C , C++ & Java</title><description><p><span style="font-size: medium;">The details on each of these programming languages are provided below:</span></p><p><span style="font-size: medium;">1. C:</span></p><p></p><ul style="text-align: left;"><li><span style="font-size: medium;">Dennis Ritchie at Bell Labs developed the general-purpose computer language C in the early 1970s.</span></li><li><span style="font-size: medium;">It is renowned for being straightforward and effective and has had a big impact on many other programming languages.</span></li><li><span style="font-size: medium;">System programming, creating operating systems, creating embedded systems, and low-level programming are all common uses for C.</span></li><li><span style="font-size: medium;">It strikes a decent mix between low-level hardware control and high-level abstractions.</span></li></ul><div><span style="font-size: large;">2. C++:</span></div><p></p><div><div><ul style="text-align: left;"><li><span style="font-size: medium;">Bjarne Stroustrup created the C++ programming language in the 1980s as an extension of the C language.</span></li><li><span style="font-size: medium;">It covers classes and objects from object-oriented programming (OOP), as well as extra capabilities like templates and exception handling.</span></li><li><span style="font-size: medium;">Application and software development, game creation, and systems programming all heavily utilize C++.</span></li><li><span style="font-size: medium;">It is appropriate for a variety of tasks because it combines low-level memory control with high-level abstractions.</span></li></ul><div><span style="font-size: medium;">3. Java:</span></div></div></div><div><div><ul style="text-align: left;"><li><span style="font-size: medium;">Midway through the 1990s, James Gosling and his colleagues at Sun Microsystems (now owned by Oracle) developed Java.</span></li><li><span style="font-size: medium;">It is intended to be platform-independent, so Java code can be used on any system that supports the Java Virtual Machine (JVM).</span></li><li><span style="font-size: medium;">Because of its "write once, run anywhere" feature, Java is well-liked for creating server-side programs, mobile apps (Android), and other things.</span></li><li><span style="font-size: medium;">It prioritizes safety and security with features like autonomous memory management (garbage collection) and upholds strict object-oriented concepts.</span></li></ul><div><span style="font-size: medium;">Each of these languages has a robust ecosystem that includes frameworks, libraries, and development tools, which adds to their adaptability and popularity across a range of industries. Please feel free to inquire if you have any special queries or would like to learn more about any specific topic relating to these languages.</span></div></div></div><div><br /></div></description><link>https://www.knowitanything.com/2023/08/c-c-and-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-5495889681183632738</guid><pubDate>Wed, 16 Aug 2023 05:18:00 +0000</pubDate><atom:updated>2023-08-16T10:48:59.569+05:30</atom:updated><title>Analog to Digital Conversion</title><description><p><span style="font-size: medium;">Electronics and digital signal processing both rely on the Analog-to-Digital Conversion (ADC) process, which transforms continuous analog impulses into discrete digital representations. The majority of digital equipment, including computers and microcontrollers, work with discrete digital values, although real-world signals (including sound, temperature, and voltage) are typically continuous in nature. For this reason, a conversion is required.</span></p><p><span style="font-size: medium;">Analog to digital conversion can be divided into many crucial steps:</span></p><p><span style="font-size: medium;">1. Sampling: The first phase entails sampling the analog signal at predetermined time intervals. A sequence of discrete voltage readings are produced when discrete points on the continuous analog signal are monitored.</span></p><p><span style="font-size: medium;">2. Quantization: Quantization is the process of giving each analog sample a digital value, often a binary value. This phase simply entails creating a finite set of discrete levels from the potential analog value range. The accuracy of the digital depiction is directly impacted by the level count. Higher precision is a result of more levels, but more bits are needed to express them.</span></p><p><span style="font-size: medium;">3. Encoding: The quantized values are now encoded into a digital format in this stage. A binary code, often in the form of binary digits (bits), is used to represent each quantized value.</span></p><p><span style="font-size: medium;">The sampling rate, the quantity of quantization levels (also known as bit depth or resolution), and the encoding technique all affect the quality of the digital representation. Higher sampling rates and bit depths produce digital representations of the original analog signal that are more precise, but they also call for more processing power and storage space.</span></p><p><span style="font-size: medium;">There are numerous ways to execute ADC, and they can be divided into various types according to their underlying principles:</span></p><p><span style="font-size: medium;">1. Successive Approximation ADC:&nbsp;This technique involves comparing the analog signal's value to a range's midpoint and making a binary choice to iteratively approximate the signal's value before refining the approximation.</span></p><p><span style="font-size: medium;">2. Delta-Sigma ADC:&nbsp;Using a feedback loop to fine-tune the digital output after oversampling the analog signal, the Delta-Sigma ADC method produces great resolution and accuracy.</span></p><p><span style="font-size: medium;">3. Flash ADC: Also referred to as parallel ADC, this technique employs a bank of comparators to identify which of a number of voltage ranges the input signal belongs to. While quick, it can also be complicated and power-hungry.</span></p><p><span style="font-size: medium;">4. Pipeline ADC: The conversion process is divided into numerous steps in a pipeline ADC design, with each stage contributing to the final digital output. For high-speed applications, it works well.</span></p><p><span style="font-size: medium;">Many different applications, including as audio processing, data gathering, sensor interfacing, telecommunications, and more, use analog-to-digital conversion to analyze and alter real-world signals in the digital realm.</span></p><p><span style="font-size: medium;">Let's explore some of the crucial facets of analog-to-digital conversion in more detail:</span></p><p><span style="font-size: medium;">Resolution and Bit Depth:&nbsp;</span></p><p><span style="font-size: medium;">An ADC's bit depth determines its resolution. The amount of bits utilized to represent each sample of the analog signal is referred to as bit depth. Greater precision is possible thanks to higher bit depth, which also makes it possible for more discrete quantization levels. For instance, a 16-bit ADC may represent the analog signal with 65,536 (216) levels while an 8-bit ADC can do it with 256 (28) discrete levels. When dealing with minute or subtle changes in the analog signal, higher resolution is especially crucial since it minimizes quantization mistakes and signal distortion.</span></p><p><span style="font-size: medium;">Sampling Rate:</span></p><p><span style="font-size: medium;">The number of samples taken per unit of time is known as the sampling rate. It is often expressed in samples per second (also known as Hertz, Hz). The Nyquist-Shannon sampling theorem states that the sample rate must be at least twice the frequency of the signal's highest frequency component in order to accurately capture the details of an analog signal during conversion. By doing this, aliasing—a distortion that can happen when high-frequency components are not correctly captured—is avoided.</span></p><p><span style="font-size: medium;">Aliasing:</span></p><p><span style="font-size: medium;">The sampling rate is defined as the number of samples taken per unit of time. It is frequently stated in samples per second, or Hz (sometimes spelled Hertz). According to the Nyquist-Shannon sampling theorem, in order to faithfully preserve the characteristics of an analog signal during conversion, the sample rate must be at least twice the frequency of the signal's highest frequency component. This prevents the distortion known as aliasing, which can occur when high-frequency components are improperly caught.</span></p><p><span style="font-size: medium;">Quantization:</span></p><p><span style="font-size: medium;">The disparity between the actual analog signal value and the quantized digital representation is known as a quantization error. Because there are only so many quantization levels, there is a certain distortion that happens during the conversion process. A higher bit depth (more quantization levels) results in lower quantization error and a more accurate approximation of the original signal.</span></p><p><span style="font-size: medium;">ADC Types:&nbsp;</span></p><p><span style="font-size: medium;">In addition to the preceding categories, there are additional specialized ADCs created for certain applications:</span></p><p></p><ul style="text-align: left;"><li><span style="font-size: medium;">Ramp ADC: This technique compares an input analog signal to a linear ramp signal.</span></li><li><span style="font-size: medium;">Integrating ADC: Integrates the input voltage over a predetermined time period to convert it.</span></li><li><span style="font-size: medium;">Dual-Slope ADC: Compares the integration times of the input signal with a known reference signal after integrating both signals.</span></li><li><span style="font-size: medium;">Pipelined ADC: Pipelined ADC that is pipelined divides the conversion into stages, each of which handles a specific piece.</span></li></ul><div><span style="font-size: medium;">Digital signal processing (DSP):&nbsp;</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">The analog signal can be treated using a variety of DSP techniques after it has been transformed to a digital representation. Filtering, modulation, noise reduction, compression, and other techniques are included. The signal can be precisely controlled and modified by digital processing, improving signal quality and adding new functionalities.</span></div><div><span style="font-size: medium;"><br /></span></div><p></p><div><span style="font-size: medium;">Applications:&nbsp;</span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">ADCs have a wide range of uses, including voice and data transmission in telecommunications, audio systems (microphones, speakers), patient monitoring in medical devices, sensor data acquisition in industrial automation, and data collection from experiments and sensors in scientific research.</span></div><div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">In conclusion, analog-to-digital conversion is a critical step that enables the processing, storing, and manipulation of real-world analog signals in the digital domain. In order to produce accurate and insightful representations of the original analog data, it requires careful consideration of elements including resolution, sampling rate, and quantization error.</span></div></div></description><link>https://www.knowitanything.com/2023/08/analog-to-digital-conversion.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-2232122805266966650</guid><pubDate>Wed, 16 Aug 2023 04:55:00 +0000</pubDate><atom:updated>2023-08-16T10:27:20.452+05:30</atom:updated><title> Passive Filters: Low-Pass, High-Pass, Band-Pass, Band-Stop</title><description><p><span style="font-size: medium;">Electronic circuits called passive filters solely use passive parts like resistors, capacitors, and inductors to filter out or let some frequencies of an input signal pass while attenuating others. They are frequently utilized in many different applications, such as communication systems, audio systems, and signal processing. Low-pass, high-pass, band-pass, and band-stop filters are the four primary categories of passive filters.</span></p><p><span style="font-size: medium;">Low-Pass Filter (LPF):&nbsp;</span></p><p><span style="font-size: medium;">A low-pass filter attenuates high-frequency components of a signal while allowing low-frequency components to flow through. It is frequently used to filter out undesired high-frequency signals or background noise from an input. The cutoff frequency marks the beginning of the signal attenuation caused by the filter's response. The attenuation of the filter is relatively modest below the cutoff frequency and sharply increases above it.</span></p><p><span style="font-size: medium;">High-Pass Filter:</span></p><p><span style="font-size: medium;">A high-pass filter (HPF) performs the inverse of what a low-pass filter does. Low-frequency components are attenuated whereas high-frequency components can get through. High-pass filters are frequently employed to eliminate undesirable low-frequency noise or to highlight a signal's higher-frequency constituents.</span></p><p><span style="font-size: medium;">Band-Pass Filter (BPF):&nbsp;</span></p><p><span style="font-size: medium;">A band-pass filter attenuates frequencies outside of its passband while permitting a certain range of frequencies, known as the passband, to pass through. When you want to isolate a certain frequency range from a signal while rejecting frequencies above and below that range, you employ band-pass filters.</span></p><p><span style="font-size: medium;">Band-Stop filter (BSF) or&nbsp; Notch filter:</span></p><p><span style="font-size: medium;">A notch filter, often referred to as a band-stop filter, allows frequencies outside of a given frequency range to pass through while rejecting frequencies inside that range. This kind of filter is frequently employed to eliminate a particular interfering frequency or to weaken a constrained frequency range in the signal.</span></p><p><span style="font-size: medium;"><br /></span></p><p><span style="font-size: medium;">The placement of passive elements (resistors, capacitors, and inductors) in the circuit affects how these filters behave. The filter's cutoff frequencies, attenuation levels, and other properties are determined by the values chosen for the component parts and how they are arranged. The frequency response roll-off's steepness is influenced by a filter's order, which is determined by the quantity of reactive elements in the filter's circuit.</span></p><p><span style="font-size: medium;">Passive filters are useful and straightforward, but they can have drawbacks, such as finite roll-off rates and sensitivity to component tolerances. To get around these restrictions and have more exact control over the filter's behavior, active filters (those that contain active elements like operational amplifiers) are sometimes utilized.</span></p></description><link>https://www.knowitanything.com/2023/08/electronic-circuits-called-passive.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-5416746546267838525</guid><pubDate>Wed, 16 Aug 2023 04:43:00 +0000</pubDate><atom:updated>2023-08-16T10:26:49.130+05:30</atom:updated><title>Oscillators </title><description><p><span style="font-size: medium;">A machine, circuit, or system that repeatedly changes or fluctuates a physical quantity over time is an oscillator. The waveforms of these variations are frequently sine waves, square waves, or sawtooth waves. Oscillators are frequently used to generate signals for a variety of applications in many disciplines, including electronics, physics, communications, and engineering.</span></p><p><span style="font-size: medium;"><span>Positive feedback is the core idea of an oscillator. When a portion of the output signal is transmitted back to the input with an amplification factor greater than 1, positive feedback happens. As a result, the system amplifies the initial input signal, resulting in long-lasting oscillations.</span>&nbsp;</span></p><p><span style="font-size: medium;">An overview of how an electronic oscillator functions is provided below:</span></p><p><span style="font-size: medium;">1. Feedback Loop: An oscillator circuit is made up of a feedback loop that includes an amplifying component (such as a transistor or operational amplifier), a network of resistors and capacitors that determines frequency, and some kind of nonlinearity to maintain oscillations.</span></p><p><span style="font-size: medium;">2. Initial Perturbation: The circuit is initially disrupted, which refers to the addition of a small amount of energy to the system. Any initial disturbance can cause this perturbation, including noise, startup mechanisms, and others.</span></p><p><span style="font-size: medium;">3. Feedback and Amplification: The circuit's amplifying component amplifies the initial perturbation signal. The feedback network then feeds this signal back into the system after it has been amplified.</span></p><p><span style="font-size: medium;">4. Phase Shift: The feedback network frequently causes the signal to phase shift by 180 degrees or multiples of that. This phase shift ensures that the signal stays in phase with the input signal after one full cycle, which is essential for maintaining oscillations.</span></p><p><span style="font-size: medium;">5. Positive Feedback: Due to positive feedback, the fed-back signal amplifies the original signal even more.</span></p><p><span style="font-size: medium;">6. Oscillation: The signal continues to be amplified and phase-shifted as it passes through the loop repeatedly. An oscillating waveform is produced as a result of the signal's constant reinforcement.</span></p><p><span style="font-size: medium;">7. Signal Output: The oscillator's output is normally obtained through coupling from a location in the circuit where the signal is at its maximum amplitude</span></p><p><span style="font-size: medium;">By altering the elements and their arrangements within the feedback loop, many types of oscillators can be produced. Typical oscillator types include:</span></p><p></p><ul style="text-align: left;"><li><span style="font-size: medium;">LC Oscillators: Inductors (L) and capacitors (C) are used in the feedback network of LC oscillators. Hartley and Colpitts oscillators are two examples.</span></li><li><span style="font-size: medium;">RC oscillators: They employ capacitors and resistors in their feedback networks. The Wien bridge oscillator and the phase-shift oscillator are two examples.</span></li><li><span style="font-size: medium;">Crystal Oscillators: Quartz crystals are used in crystal oscillators to set the frequency. They are frequently employed in electrical devices including clocks, radios, and microcontrollers and offer great stability and accuracy.</span></li><li><span style="font-size: medium;">Voltage-Controlled Oscillators (VCOs): These oscillators enable an external voltage to regulate the output waveform's frequency.</span></li></ul><p></p><ul></ul><span style="font-size: medium;">Oscillators are essential in a wide range of applications, including the creation of carrier waves for communication systems, clock signals for digital circuits, test signals for instrumentation, and timing references for a variety of electronic devices.</span></description><link>https://www.knowitanything.com/2023/08/oscillators-explanation.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-6105397900624020257</guid><pubDate>Tue, 15 Aug 2023 09:30:00 +0000</pubDate><atom:updated>2023-08-15T15:01:16.118+05:30</atom:updated><title>Laplace Transform</title><description><p>A mathematical method called the Laplace transform is used to study and resolve linear time-invariant differential equations. It is an effective tool for the study of systems with continuous-time dynamics in the fields of mathematics, engineering, physics, and many others.</p><div><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; margin: 1.25em 0px; white-space-collapse: preserve;"><span style="background-color: white;">The Laplace transform is defined for a function <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; 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--tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; 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--tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> defined for <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.7719em; vertical-align: -0.136em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">≥</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6444em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span></span></span> as follows:</span></p><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; margin: 1.25em 0px; white-space-collapse: preserve;"><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; background-color: white; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; 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--tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Caligraphic;">L</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">{</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)}</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.2151em; vertical-align: -0.3558em;"></span><span class="mop" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mop op-symbol small-op" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Size1; margin-right: 0.19445em; position: relative; top: -0.0006em;">∫</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t vlist-t2" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; margin-right: -2px; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8593em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-left: -0.1945em; margin-right: 0.05em; position: relative; top: -2.3442em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.2579em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">∞</span></span></span></span><span class="vlist-s" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; font-size: 1px; min-width: 2px; vertical-align: bottom; width: 2px;"></span></span><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; 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--tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">e</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.7936em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">d</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span></span></span></span></span></p></div><div><div>where s is an intricate number. The new function of the complex variable s, written as F(s), is the outcome of applying the Laplace transform to a function.</div><div><br /></div><div>For the purpose of resolving differential equations and analyzing systems, the Laplace transform has numerous qualities that are advantageous:</div></div><div><ol style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-reset: list-number 0; display: flex; flex-direction: column; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; list-style: none; margin: 1.25em 0px; padding: 0px; white-space-collapse: preserve;"><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span style="background-color: white;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600;">Linearity</span>: <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">�</mi><mo stretchy="false">{</mo><mi>�</mi><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>+</mo><mi>�</mi><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mi>�</mi><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>+</mo><mi>�</mi><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Caligraphic;">L</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">{</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">a</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">b</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.03588em;">g</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)}</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">a</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">+</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">b</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">G</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span>, where <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.4306em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">a</span></span></span></span></span> and <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 0.6944em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">b</span></span></span></span></span> are constants.</span></p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span style="background-color: white;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600;">Shifting Theorem</span>: <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">�</mi><mo stretchy="false">{</mo><msup><mi>�</mi><mrow><mi>�</mi><mi>�</mi></mrow></msup><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo>−</mo><mi>�</mi><mo stretchy="false">)</mo></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0436em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Caligraphic;">L</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">{</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">e</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.7936em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">a</span><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)}</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; 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--tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">a</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span>, which allows shifting a function in the time domain to multiplication by an exponential term in the Laplace domain.</span></p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span style="background-color: white;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600;">Derivative Theorem</span>: <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; 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box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0019em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; 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border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">′</span></span></span></span></span></span></span></span></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)}</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span><span class="mbin" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2222em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span>, where <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span> is the initial value of the function.</span></p></li><li style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; counter-increment: list-number 1; display: block; margin-bottom: 0px; margin-top: 0px; min-height: 28px; padding-left: 0.375em;"><p style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; margin: 0px;"><span style="background-color: white;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-weight: 600;">Integral Theorem</span>: <span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: KaTeX_Main, &quot;Times New Roman&quot;, serif; font-feature-settings: normal; font-kerning: auto; font-optical-sizing: auto; font-size: 1.21em; font-stretch: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-variation-settings: normal; line-height: 1.2; text-rendering: auto;"><span class="katex-mathml" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">�</mi><mo stretchy="false">{</mo><msubsup><mo>∫</mo><mn>0</mn><mi>�</mi></msubsup><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>�</mi><mi>�</mi><mo stretchy="false">}</mo><mo>=</mo><mfrac><mrow><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo></mrow><mi>�</mi></mfrac></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.3443em; vertical-align: -0.3558em;"></span><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Caligraphic;">L</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">{</span><span class="mop" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mop op-symbol small-op" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Size1; margin-right: 0.19445em; position: relative; top: -0.0006em;">∫</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t vlist-t2" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; margin-right: -2px; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.9885em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-left: -0.1945em; margin-right: 0.05em; position: relative; top: -2.3442em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">0</span></span></span><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.2579em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">t</span></span></span></span><span class="vlist-s" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; font-size: 1px; min-width: 2px; vertical-align: bottom; width: 2px;"></span></span><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.3558em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative;"></span></span></span></span></span></span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.1132em;">τ</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.1667em;"></span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">d</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.1132em;">τ</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">}</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span><span class="mrel" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">=</span><span class="mspace" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; margin-right: 0.2778em;"></span></span><span class="base" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.355em; vertical-align: -0.345em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mopen nulldelimiter" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; width: 0.12em;"></span><span class="mfrac" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t vlist-t2" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; margin-right: -2px; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row; text-align: center;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 1.01em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative; top: -2.655em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 3em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span></span></span></span><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative; top: -3.23em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 3em; overflow: hidden; width: 0px;"></span><span class="frac-line" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-color: currentcolor; border-image: initial; border-style: solid; border-width: 0px 0px 0.04em; box-sizing: border-box; display: inline-block; min-height: 1px; width: 27.5156px;"></span></span><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative; top: -3.485em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 3em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)</span></span></span></span></span><span class="vlist-s" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; font-size: 1px; min-width: 2px; vertical-align: bottom; width: 2px;"></span></span><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row; text-align: center;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.345em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; position: relative;"></span></span></span></span></span><span class="mclose nulldelimiter" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; width: 0.12em;"></span></span></span></span></span></span>, which relates the Laplace transform of an integral to the Laplace transform of the original function.</span></p></li></ol><div><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">Differential equations can be converted into algebraic equations in the Laplace domain using the Laplace transform, which are frequently simpler to solve. You can find the inverse Laplace transform to get the answer in the time domain after resolving the changed variable.</span></span></div></div><div><span style="background-color: white;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; white-space-collapse: preserve;"><br /></span></span></div><div><span style="background-color: white;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; white-space-collapse: preserve;">The inverse Laplace transform is denoted as </span><span class="math math-inline" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; white-space-collapse: preserve;"><span class="katex" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; 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--tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px; box-sizing: border-box; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi mathvariant="script">�</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">{</mo><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></semantics></math></span><span aria-hidden="true" class="katex-html" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; 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box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.10764em;">f</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; 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border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; position: relative; white-space: nowrap; width: min-content;"><span class="strut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mathcal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Caligraphic;">L</span><span class="msupsub" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="vlist-t" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border-collapse: collapse; border: 0px solid currentcolor; box-sizing: border-box; display: inline-table; table-layout: fixed;"><span class="vlist-r" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-row;"><span class="vlist" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: table-cell; height: 0.8141em; position: relative; vertical-align: bottom;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: block; height: 0px; margin-right: 0.05em; position: relative; top: -3.063em;"><span class="pstrut" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; height: 2.7em; overflow: hidden; width: 0px;"></span><span class="sizing reset-size6 size3 mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; display: inline-block; font-size: 0.7em;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;"><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">−</span><span class="mord mtight" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">1</span></span></span></span></span></span></span></span></span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">{</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic; margin-right: 0.13889em;">F</span><span class="mopen" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">(</span><span class="mord mathnormal" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box; font-family: KaTeX_Math; font-style: italic;">s</span><span class="mclose" style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid currentcolor; box-sizing: border-box;">)}</span></span></span></span></span><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; font-size: 16px; white-space-collapse: preserve;">, and it's the process of finding the original function from its Laplace transform.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">The Laplace transform is frequently used in control theory, signal processing, circuit analysis, and other areas where differential equations play an important part. It is an excellent tool for studying and solving linear time-invariant systems.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Let's explore some other ideas and uses for Laplace transforms in more detail: For all complex values of s, the Laplace transform might not converge in the region of convergence (ROC). The Region of Convergence (ROC) is the collection of s values for which the integral defining the Laplace transform converges. When working with inverse Laplace transforms, the ROC is a crucial factor to take into account because it determines the range of values for which the inverse transform is valid.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Complex Frequency Domain Analysis: </span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">The Laplace transform broadens the definition of frequency analysis to include the complex plane in addition to the more conventional real-valued frequencies (as in Fourier analysis). The imaginary part of the complex s shows oscillatory activity, whereas the real part represents exponential growth or decay. When working with systems that display transient and steady-state responses, this complicated frequency domain analysis is especially helpful.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Initial Value Theorem and Final Value Theorem:</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Theorems of initial value and final value shed light on how a system behaves under initial conditions and under steady-state conditions as time approaches zero and infinity, respectively. The Final Value Theorem connects the steady-state behavior to the value of the function's Laplace transform as s approaches zero, whereas the Initial Value Theorem connects the initial condition of a function to the value of its Laplace transform at &nbsp;s=∞.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Partial Fraction Decomposition:</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Decomposing a complex rational function in the Laplace domain into a sum of smaller terms is frequently necessary when working with complex Laplace transforms. Partial fraction decomposition is a technique for simplifying the algebraic manipulation and solution of Laplace-transformed equations.</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><br /></span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Application areas:</span></span></div><div><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Laplace transforms are used in a variety of domains, including:</span></span></div><div><ul style="text-align: left;"><li><span style="background-color: white; white-space-collapse: preserve;"><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;">Control Systems: </span></span><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">Laplace transforms are essential for studying and creating control systems because they make it easier to comprehend how the system reacts to various inputs and to create controllers.</span></span></li><li><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">Circuit Analysis: Laplace transforms make it easier to analyze linear electrical circuits and to analyze transient and steady-state responses.</span></span></li><li>Control Systems: Laplace transforms are crucial for analyzing and designing control systems, where they help in understanding the system's response to different inputs and in designing controllers.</li><li>Circuit Analysis: Laplace transforms simplify the analysis of linear electrical circuits, aiding in the analysis of transient and steady-state responses.</li><li>Mechanical Systems:&nbsp;Laplace transforms make it easier to analyze linear electrical circuits and to analyze transient and steady-state responses.</li><li><span style="white-space-collapse: preserve;">Signal Processing: Laplace transforms are used in signal processing to analyze and filter continuous-time signals.</span></li><li><span style="white-space-collapse: preserve;">Differential Equations: Laplace transforms offer a distinct method for resolving ordinary and partial differential equations that might appear in a variety of scientific and engineering contexts.</span></li><li><span style="background-color: white;"><span style="--tw-border-spacing-x: 0; --tw-border-spacing-y: 0; --tw-ring-color: rgba(69,89,164,.5); --tw-ring-offset-color: #fff; --tw-ring-offset-shadow: 0 0 transparent; --tw-ring-offset-width: 0px; --tw-ring-shadow: 0 0 transparent; --tw-rotate: 0; --tw-scale-x: 1; --tw-scale-y: 1; --tw-scroll-snap-strictness: proximity; --tw-shadow-colored: 0 0 transparent; --tw-shadow: 0 0 transparent; --tw-skew-x: 0; --tw-skew-y: 0; --tw-translate-x: 0; --tw-translate-y: 0; border: 0px solid rgb(217, 217, 227); box-sizing: border-box; color: var(--tw-prose-bold); font-size: 16px; white-space-collapse: preserve;">Fluid Dynamics and Heat Transfer</span><span style="color: #374151; font-size: 16px; white-space-collapse: preserve;">: </span></span><span style="white-space-collapse: preserve;">Laplace transforms can be used in engineering applications to solve differential equations relating to fluid dynamics and heat transport.</span></li></ul><div><span style="white-space-collapse: preserve;">Inverse Laplace Transform Techniques:</span></div></div><div><span style="white-space-collapse: preserve;"><div>Techniques for the Inverse Laplace Transform: Finding the Inverse Laplace Transform can be difficult, particularly for complex functions. Partial fraction expansion, contour integration, the convolution theorem, and employing tables of Laplace transforms are methods for locating inverse Laplace transforms.</div><div><br /></div><div>Overall, the Laplace transform is a comprehensive mathematical tool that allows the analysis and solution of a wide variety of issues in engineering, physics, and other scientific fields by bridging the gap between differential equations and algebraic equations.</div></span></div></description><link>https://www.knowitanything.com/2023/08/a-mathematical-method-called-laplace.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7893331170857731505</guid><pubDate>Tue, 15 Aug 2023 09:03:00 +0000</pubDate><atom:updated>2023-08-15T14:33:14.869+05:30</atom:updated><title>Fourier Transform</title><description><p>In order to examine functions and signals in terms of their frequency components, the Fourier Transform is a mathematical approach used in both signal processing and mathematics. It enables you to describe a function or signal as the accumulation of various-frequency sinusoidal waves. The French mathematician and physicist Jean-Baptiste Joseph Fourier is honored with the transform's name.</p><p>Both continuous and discrete functions can use the Fourier Transform. The continuous Fourier Transform (CFT) and the discrete Fourier Transform (DFT) are the two primary variants of the transform.</p><p>Continuous Fourier Transform (CFT):</p><p>For a continuous function f(t), the CFT is defined as:</p><p>F(ω) = ∫[from -∞ to +∞] f(t) * e^(-iωt) dt,</p><p>where F(ω) is the complex-valued function representing the frequency components of the original function f(t), ω is the angular frequency, i is the imaginary unit, and t is time</p><p>Discrete Fourier Transform (DFT):</p><p>For a discrete function f[n] sampled at discrete points in time, the DFT is defined as:</p><p>F[k] = Σ[from n = 0 to N-1] f[n] * e^(-i2πkn/N),</p><p>where F[k] represents the frequency components of the discrete function, N is the number of samples, k is the index of the frequency component, and n is the index of the sample.</p><p>The frequency-domain representation can be transformed back into the time-domain representation using the inverse transforms, which are also available for the continuous and discrete situations.</p><p>Numerous industries, including signal processing, image analysis, audio processing, communications, physics, and engineering, use the Fourier Transform extensively. Filtering, noise reduction, compression, pattern recognition, and other processes are among its uses. A popular method for quickly calculating the DFT is the Fast Fourier Transform (FFT) algorithm, particularly in applications involving digital signal processing.</p><p>The Fourier Transform is a key component of contemporary data analysis and manipulation because it is a potent tool for examining the frequency content of signals and functions.</p><p>Let's explore some fundamental ideas about the Fourier Transform and its uses in more detail:</p><p>Frequency Components and Spectra:&nbsp;</p><p>The Fourier Transform is used to convert a signal from the time domain to the frequency domain. This process displays the various sinusoidal components that make up the signal. The complex values in the modified domain serve as a representation of these elements. Each frequency component's amplitude is represented by the magnitude of these complex numbers, and each component's relative phase shift is shown by the phase angle.</p><p>Convolution Theorem:&nbsp;</p><p>The Convolution Theorem is one of the core characteristics of the Fourier Transform. According to this, multiplication in the frequency domain corresponds to convolution in the time domain, and vice versa. The ability to transform signals into the frequency domain, conduct operations on them, and then translate them back to the time domain makes this capability very helpful in signal processing applications.</p><p>Windowing and Leakage:&nbsp;</p><p>Windowing is frequently utilized in real-world Fourier Transform applications, particularly with discrete signals. To restrict the area of the signal being studied, this involves multiplying the signal by a window function. Windowing, however, can result in "spectral leakage," where the frequency components of the signal spread out as a result of the window's limited length. The precision of frequency analysis may be impacted by this.</p><p>Power Spectral Density (PSD):&nbsp;</p><p>The PSD is a metric for determining how a signal's power is divided among various frequency components. It offers information about the relative potency of different frequencies that are present in the signal. The PSD is frequently used to examine a signal's frequency content and pinpoint the major frequency components.</p><p>Applications:&nbsp;</p><p>1. Signal Processing: Fourier Transforms are widely employed in signal processing for feature extraction, denoising, and filtering. They are applied in telecommunications, audio processing, and picture analysis, among other industries.</p><p>2. Communication Systems: Signals are modulated onto carrier frequencies in telecommunications. The original message signal can be recovered and the modulation process is analyzed using the Fourier Transform.</p><p>3. Image processing: The Fourier Transform is used in image analysis to examine image textures and extract information about spatial frequency. In compression techniques like JPEG, it is also utilized.</p><p>4. Spectroscopy: Fourier Transforms are used in various forms of spectroscopy in research and engineering to examine signals from diverse light or other radiation wavelengths.</p><p>5. Quantum Mechanics: Fourier Transforms are frequently used to represent the wavefunctions in quantum mechanics, allowing for the investigation of particle behavior in various momentum and position states.</p><p>6. Vibration Analysis: Fourier Transforms are used in mechanical engineering to analyse the frequency components of machinery and structures and to assess vibrations.</p><p>Extensions and Variations:&nbsp;</p><p>1. Short-Time Fourier Transform (STFT): It is appropriate for non-stationary signals because it enables analysis of how a signal's frequency components fluctuate over time.</p><p>2. Wavelet Transform: The Wavelet Transform employs functions that are localized in both time and frequency, as opposed to the Fourier Transform's fixed sinusoidal basis, making it appropriate for evaluating signals with a range of frequencies.</p><p>3. Spectral Analysis: Statistical methods like the Periodogram and the Welch Method are used to estimate power spectral densities from finite-length data in addition to Fourier Transforms.</p><p>These are merely a few further uses for and facets of the Fourier Transform. It continues to be a crucial instrument in a wide range of scientific and technical disciplines due to its accessibility and adaptability in comprehending the frequency content of signals and functions.</p><div><br /></div></description><link>https://www.knowitanything.com/2023/08/fourier-transform.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-8466452817126733420</guid><pubDate>Tue, 15 Aug 2023 05:28:00 +0000</pubDate><atom:updated>2023-08-15T10:58:54.157+05:30</atom:updated><title>MOSFET and JFET</title><description><p>Two further types of field-effect transistors that are frequently used in electronics for a variety of applications are MOSFETs and junction field-effect transistors (JFETs). They function differently from BJTs but are still utilized for amplification, switching, and signal processing. Let's examine each of these in more detail:</p><p>Metal-oxide semiconductor field-effect transistor (MOSFET):</p><p>A gate, a source, and a drain are the three terminals of a MOSFET. Due to its high input impedance, low power consumption, and capacity to switch huge currents with little power loss, they are widely utilized. The enhancement-mode MOSFET and the depletion-mode MOSFET are the two primary varieties of MOSFETs.</p><p></p><ul style="text-align: left;"><li>Enhancement-Mode MOSFET: The flow of current between the source and drain terminals is regulated by a voltage given to the enhancement-mode MOSFET's gate terminal. Current can flow between the source and drain when the gate-source voltage is strong enough to form a "inversion layer" in the semiconductor.</li><li>Depletion-Mode MOSFET: A depletion-mode MOSFET typically conducts current when no voltage is provided to the gate terminal, in contrast to an enhancement-mode MOSFET. When a negative gate-source voltage is used, the conductivity is decreased and the current flow is managed.</li></ul><p></p><p>MOSFETs are frequently used in power electronics, amplifiers, and digital integrated circuits (CMOS technology).</p><p>JFET (Junction Field-Effect Transistor):</p><p>The three-terminal JFET (Junction Field-Effect Transistor) has a gate, a source, and a drain. They generally function in low-power, high-impedance applications and, in terms of voltage amplification, resemble vacuum tubes. N-channel JFETs and P-channel JFETs are the two primary varieties of JFETs.</p><p></p><ul style="text-align: left;"><li>N-channel JFET: In an N-channel JFET, the voltage applied to the gate regulates the current flowing between the source and drain. With respect to the source, applying a negative voltage to the gate causes the current flow to increase (depletion mode), whilst applying a positive voltage causes the current flow to decrease (enhancement mode).</li><li>P-channel JFET: A JFET with a P-channel controls the current flow in the opposite way. With respect to the source, a positive voltage applied to the gate causes the current flow to increase (depletion mode), whereas a negative voltage causes the current flow to decrease (enhancement mode).</li></ul><p></p><p>JFETs are frequently employed in audio equipment's low-noise amplifiers, impedance converters, and high-impedance input stages.</p><p>In conclusion, field-effect transistors such as MOSFETs and JFETs use an electric field produced by a voltage applied to the gate terminal to regulate the current flowing between the source and drain terminals. They are chosen depending on the particular requirements of an application and each has unique qualities.</p></description><link>https://www.knowitanything.com/2023/08/mosfet-and-jfet.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-1162576734369074060</guid><pubDate>Tue, 15 Aug 2023 05:19:00 +0000</pubDate><atom:updated>2023-08-15T10:49:11.399+05:30</atom:updated><title>Bipolar Junction Transistor (BJT's)</title><description><p>Bipolar Junction Transistors (BJTs) are semiconductor three-terminal devices utilized in a variety of electronic applications, including signal modulation, switching, and amplification. NPN (Negative-Positive-Negative) and PNP (Positive-Negative-Positive) are the two primary categories of BJTs. The placement of their semiconductor layers and the way current flows differ between these two types. An overview of NPN and PNP transistors is given below:</p><p>1. NPN Transistor:</p><p></p><ul style="text-align: left;"><li>Structure: A layer of P-type semiconductor material (positive) is sandwiched between two layers of N-type semiconductor material (negative) in an NPN transistor. The N-type layers serve as the emitter and collector, with the P-type layer serving as the base.</li><li>Current Flow: In an NPN transistor, the base functions as a control terminal to regulate the current flow, which travels from the emitter to the collector.</li><li>Amplification: NPN transistors are frequently employed in amplifying applications. A signal is amplified when a tiny current enters the base and controls a bigger current that is flowing from the emitter to the collector.</li></ul><div>2. PNP Transistor:</div><p></p><div><ul style="text-align: left;"><li>Structure: A layer of N-type semiconductor material (negative) is sandwiched between two layers of P-type semiconductor material (positive) in a PNP transistor. The base is an N-type layer, whereas the emitter and collector are P-type layers.</li><li>Current Flow: In a PNP transistor, the base regulates the current flow from the collector to the emitter.</li><li>Amplification: Similar to NPN transistors, PNP transistors can also be used for amplification. A tiny current at the base regulates the flow of current from the collector to the emitter.</li></ul><div>Transistor Amplifier Configurations:&nbsp;</div><div><br /></div><div>Transistors are frequently employed as amplifiers to strengthen a weak input signal. The common emitter (CE), common base (CB), and common collector (CC) configurations are the three fundamental types of transistor amplifiers.</div><div><br /></div><div>1. Common Emitter (CE) Configuration:</div><div><ul style="text-align: left;"><li>The emitter terminal is shared by the input and output circuits in a CE setup.</li><li>High voltage gain and moderate current gain are provided by CE amplifiers.</li><li>The base-emitter junction receives the input, while the collector-emitter junction receives the output.</li><li>Phase inversion exists between the input and output of CE amplifiers.</li></ul><div>2. Common Base (CB) Configuration:</div></div></div><div><ul style="text-align: left;"><li>The base terminal is shared by the input and output circuits in a CB design.</li><li>High current gain and unity voltage gain (i.e., no voltage amplification) are provided by CB amplifiers.</li><li>The emitter-base junction receives the input, while the collector-base junction receives the output.</li><li>Low input impedance and high output impedance characterize CB amplifiers.</li></ul><div>3. Common Collector (CC) Configuration (Emitter Follower):</div><div><div><ul style="text-align: left;"><li>The collector terminal is shared by the input and output circuits in a CC design.</li><li>CC amplifiers offer high voltage gain and unity current gain.</li><li>The base-emitter junction receives the input, while the collector-emitter junction receives the output.</li><li>High input impedance and low output impedance characterize CC amplifiers.</li></ul></div><div>Engineers can select the optimal amplifier layout for a given application by weighing trade-offs in terms of voltage gain, current gain, input/output impedance, and phase relationships.</div></div><div><br /><br /></div></div></description><link>https://www.knowitanything.com/2023/08/bipolar-junction-transistor-bjts.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-4475055823949789526</guid><pubDate>Tue, 15 Aug 2023 04:53:00 +0000</pubDate><atom:updated>2023-08-15T10:23:13.657+05:30</atom:updated><title>Ohm's law </title><description><p>The relationship between voltage, current, and resistance in an electrical circuit is described by the fundamental law known as Ohm's Law in electrical engineering and physics. It bears the name Georg Simon Ohm after the German physicist who developed it in the 19th century.</p><p>Ohm's Law can be formulated mathematically as:</p><p>V = I * R</p><p>Where:</p><p></p><ul style="text-align: left;"><li>Volts (V) are used to denote the voltage across a component.</li><li>I is a unit of measurement for the component's current in amperes (A).</li><li>R stands for the component's resistance in ohms ().</li></ul><p></p><p>The voltage across a resistor (or any other component with a constant resistance), in other words, is directly proportional to the current passing through it, according to Ohm's Law. As long as the temperature and other environmental variables stay constant, this relationship is still true.</p><p>A key idea in comprehending and studying electrical circuits is Ohm's Law. It serves as the basis for many facets of electronics and electrical engineering and is used to calculate unknown values, such as current or resistance, given the other two variables.</p><p>Let's examine Ohm's Law's elements and ramifications in more detail:</p><p>1. Voltage: The electrical potential difference between two locations in a circuit is known as voltage (V). It's frequently called "electric pressure." Voltage, which is measured in volts (V), is often provided by a power source like a battery or a generator.</p><p>2. Current: Electric charge flowing through a conductor, such as a wire, is known as current (I). It expresses how many electrons pass a specific point in the circuit in one unit of time and is expressed in amperes (A). Electrical circuits run on current, which also powers electronic equipment.</p><p>3. Resistance (R): Resistance is a unit of measurement for how much a component resists current flow. It is brought on by things like the material's characteristics, its size, and the temperature. Ohms () are used to measure resistance. High resistance substances are insulators, whereas low resistance substances are conductors.</p><p>4. Variations of Ohm's Law:</p><p></p><ul style="text-align: left;"><li>Ohm's Law can be modified to get the following two helpful equations:</li><li>When voltage and resistance are known, you may compute current using the equation I = V / R.</li><li>When voltage and current are known, you may calculate resistance using the equation R = V / I.</li><li>These changes aid in circuit analysis and parameter calculation.</li></ul><div>5.&nbsp;Ohmic and Non-Ohmic Components:</div><p></p><div><ul style="text-align: left;"><li>Ohmic parts precisely adhere to Ohm's Law, keeping their resistance constant no matter how much voltage or current is supplied. The majority of resistors are examples.</li><li>Ohm's Law does not apply to non-Ohmic components (such as diodes or transistors) since they experience resistance variations with voltage or current.</li></ul><div>6. Series and Parallel Circuits:</div></div><div><ul style="text-align: left;"><li>The same current passes through every component in a series circuit since they are all connected end to end. Overall resistance increases.</li><li>Components in parallel circuits have varying currents while sharing the same voltage. The sum of the inverses of each individual resistance equals the inverse of the total resistance.</li></ul><div><div>7. Effects of temperature:</div><div><ul style="text-align: left;"><li>Due to the material's temperature coefficient, resistance might fluctuate with temperature. Resistance generally increases with temperature for conductors and decreases for semiconductors.</li></ul></div><div>8.Applications:</div><div><ul style="text-align: left;"><li>Ohm's Law is fundamental to circuit design and analysis, assuring secure operation, and performance enhancement.</li><li>It is essential for power distribution, telecommunications, electronics, and other systems.</li></ul><div><div>9. Limitations:</div><div><ul style="text-align: left;"><li>Ohm's Law typically applies to linear components that are functioning within predetermined boundaries. It might not apply to complicated or highly nonlinear devices.</li></ul></div><div><br /></div><div>Overall, Ohm's Law is a foundational principle of electrical engineering that enables professionals and amateurs to comprehend and work with electrical circuits to build a variety of devices, from straightforward lightbulbs to sophisticated computer systems.</div></div></div></div></div></description><link>https://www.knowitanything.com/2023/08/ohms-law.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-1434390182776937523</guid><pubDate>Tue, 15 Aug 2023 04:34:00 +0000</pubDate><atom:updated>2023-08-15T10:04:35.366+05:30</atom:updated><title>Active and Passive Components</title><description><p>The basic building elements of electronic circuits that are utilized to modify, control, and process electrical signals are called active and passive components. Depending on their innate characteristics, they play different roles in the design and functioning of circuits. An outline of active and passive parts is provided below:</p><p>Passive Components:</p><p>Passive components are those that respond to changes in voltage and current but do not gain or amplify anything. Passive components do not need an external power source to work.</p><p>1. Resistor: A resistor is a passive part that prevents current from flowing. It is frequently used in circuits to limit current, divide voltage, and supply load.</p><p>2.Capacitor: Electrical energy is stored and released by a capacitor. It is made up of two conductive plates separated by a dielectric (insulating substance). Energy storage, signal coupling, filtering, and timing circuits all require capacitors.</p><p>3. Inductor: When current passes through an inductor, it stores energy in the form of a magnetic field. It is employed in processes like energy storage, filtration, and impedance matching because it resists changes in current.</p><p>4. Transformer: A transformer is made up of two or more inductors that have a common magnetic field. It is frequently used in power distribution and signal isolation. It is utilized to transport electrical energy across various voltage levels.</p><p>Active Components:&nbsp;</p><p>Active components can amplify, regulate, or otherwise alter electrical signals, but they also need an external power source to work.</p><p>1. Transistor:&nbsp;A transistor is an active semiconductor device that has the ability to switch and amplify electronic signals. They are widely employed in amplifiers, switches, oscillators, and other electronic devices. They are available in several forms, including bipolar junction transistors (BJT) and field-effect transistors (FET).</p><p>2. Operational Amplifier (Op-Amps): Op-Amps are flexible integrated circuits that have high gain and differential input capability. They are extensively utilized in mathematical processes, signal conditioning, filtering, and amplification.</p><div>3.&nbsp;Integrated Circuits (ICs): ICs are assemblies of both active and passive parts that come in a single package. They might incorporate memory devices, logic gates, microcontrollers, and specialized analog and digital operations.</div><div><br /></div><div>4.&nbsp;Diode: While most types of diodes are passive, forward-biased versions of certain, such as light-emitting diodes (LEDs) and laser diodes, emit light. Other active diodes that produce an electrical current in reaction to light include photodiodes and solar cells.</div><div><br /></div><div>5.Operational Transconductance Amplifier (OTA):&nbsp;These specialized switching devices with great current handling capabilities include the thyristor and the triac. Triacs are utilized for AC switching, whereas thyristors are used in situations needing high-power control.</div><div><br /></div><div>6. Thyristor and Triac: These specialized switching devices with great current handling capabilities include the thyristor and the triac. Triacs are utilized for AC switching, whereas thyristors are used in situations needing high-power control.</div><div><br /></div><div>In contrast to passive components, which are used for voltage division, current limiting, energy storage, and signal filtering, active components are essential for signal amplification, processing, and active control in circuits. To achieve desired functionality and performance, a well-designed circuit frequently combines both active and passive components.</div></description><link>https://www.knowitanything.com/2023/08/active-and-passive-components.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-531952734288443631</guid><pubDate>Tue, 15 Aug 2023 04:13:00 +0000</pubDate><atom:updated>2023-08-15T09:43:51.182+05:30</atom:updated><title>Types of amplifiers</title><description><p>Electronic devices called amplifiers boost the amplitude of an input signal while maintaining its structure and properties. There are various amplifier kinds, each of which is created for a particular application and set of performance requirements. Here are a few popular amplifier types:</p><p>Common Emitter Amplifier:&nbsp;Bipolar junction transistor (BJT) amplifier of the common emitter type boosts the voltage between the transistor's emitter and collector. It has a high voltage gain and is frequently used for general-purpose amplification.</p><p>Common Base Amplifier: The common base amplifier boosts the current flowing between the transistor's emitter and collector in a different BJT amplifier design. It is frequently employed in high-frequency and impedance matching applications.</p><div class="separator" style="clear: both; text-align: left;">Common Collector Amplifier (Emitter Follower):&nbsp;This BJT arrangement uses an emitter follower and a common collector amplifier, with the output coming from the emitter. The emitter follower is a good choice for impedance buffering since it has a high input impedance, a low output impedance, and unity voltage gain.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Differential Amplifier: Differential amplifiers boost the differences between two input signals while rejecting any common-mode signals (signals that are present in both inputs at the same time). In areas like instrumentation and signal processing, differential amplifiers are essential.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Operational Amplifier (Op-Amp): Op-amps are extremely adaptable integrated circuit amplifiers with high input impedance, low output impedance, and exceptionally high open-loop gain. Amplification, filtering, signal conditioning, and mathematical computations are just a few of the many uses for them.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both;">Power Amplifier: Power amplifiers are made to produce a great amount of output power to drive speakers or other high-power loads. They are frequently employed in RF transmitters, motor control devices, and audio systems.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Class A Amplifier: This type of amplifier has low efficiency but strong linearity since the output transistor(s) are continually conducting during operation.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Class B amplifier: They employ two transistors, each of which only conducts half of the input waveform. They can introduce crossover distortion despite being more effective than Class A.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;"><div class="separator" style="clear: both;">Class AB Amplifier: They combine characteristics of Class A and Class B amplifiers with the goal of lowering crossover distortion while retaining acceptable efficiency.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Class D Amplifier: Also referred to as a switching amplifier, Class D amplifiers maximize efficiency by utilizing pulse-width modulation. Efficiency is vital in audio applications, as they are frequently employed.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Differential amplifier: They boost the differences between two input signals while rejecting any common-mode signals (signals that are present in both inputs at the same time). In areas like instrumentation and signal processing, differential amplifiers are essential.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Instrumentation Amplifier: High input impedance and common-mode rejection are provided by this kind of differential amplifier, known as an instrumentation amplifier. It is frequently employed for precise low-level signal measurement.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Voltage Amplifier : This amplifier mainly enhances voltage signals. It comes in a variety of shapes and arrangements, like the ones described above.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">Current Amplifier: The main purpose of this amplifier is to amplify current signals. Despite being less popular than voltage amplifiers, it has its uses.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;"><div class="separator" style="clear: both;">Logarithmic Amplifier (Log Amp): Logarithmic amplifiers are used to compress the output of a wide range of input signal amplitudes. They are frequently employed in signal-compression-related applications, like audio processing.</div><div class="separator" style="clear: both;"><br /></div><div class="separator" style="clear: both;">These are only a few of the several amplifier designs that are available, each suited to certain uses and specifications. The signal characteristics, required gain, frequency range, power requirements, and the intended application are only a few examples of the variables that influence the choice of amplifier type.</div><div><br /></div></div></div></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><br /><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><br /><p><br /></p></description><link>https://www.knowitanything.com/2023/08/types-of-amplifiers.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-287601198565072039</guid><pubDate>Mon, 14 Aug 2023 11:39:00 +0000</pubDate><atom:updated>2023-08-14T17:09:14.263+05:30</atom:updated><title>Java Runtime Environment</title><description><p>The Java platform is not complete without the Java Runtime Environment (JRE). It offers the environment required to run Java programs and applets. The Java Virtual Machine (JVM), libraries, and other resources necessary for running Java programs are all included in the JRE.</p><p>Here is a list of the essential elements of the Java Runtime Environment:</p><p>Java Virtual Machine (JVM): The JRE's main component is the JVM. Java bytecode, which is the compiled version of Java source code, is executed by the Java Virtual Machine (JVM). It controls garbage collection, maintains memory, and acts as a layer of abstraction between the Java code and the underlying operating system and hardware.</p><p>Java Class Libraries: These are a collection of pre-compiled classes and methods that give Java programs access to a wide range of functionality. They cover a wide range of topics, including networking, file processing, graphical user interfaces, and fundamental data types and data structures.</p><p>Java API Documentation: The JRE comes with thorough Java Standard Library documentation that gives programmers details on the classes, methods, and packages that are accessible to their applications.</p><p>Java Compiler Tools: Tools for compiling Java source code into bytecode, such as the Java Compiler (javac), are sometimes included in JRE releases, however this isn't always the case.</p><p>Security and Runtime Environment Management: To guarantee that Java programs operate in a controlled environment, the JRE incorporates security methods. This contains functions like the security manager, which can limit the things Java code can do.</p><p><br /></p><p>It's crucial to remember that the JRE is designed to run Java programs. The Java Development Kit (JDK), which includes the JRE as well as extra tools like the Java Compiler (javac) and debugging utilities, is often used for development purposes.</p><p>The JRE is normally delivered as part of the many versions of Java that have been released throughout time. To run Java applications created with the matching version of the Java language, users must have the correct JRE version installed on their computers.</p></description><link>https://www.knowitanything.com/2023/08/java-runtime-environment.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-483555567317897196</guid><pubDate>Mon, 14 Aug 2023 11:31:00 +0000</pubDate><atom:updated>2023-08-14T17:01:06.760+05:30</atom:updated><title>ArrayList in Java</title><description><p>An ArrayList is a dynamic array-like data structure offered by the Java Collections Framework in the programming language. It is a component of the java.util package and has a number of benefits over standard arrays, including automatic resizing, simple element insertion and removal, and many utility methods. An overview of using an ArrayList in Java is given below:</p><p>1. Import the class ArrayList:</p><p>import java.util.ArrayList;</p><div>2. Create an array list:&nbsp;</div><div><br /></div><div>You can either start with an empty array list or give it some default values.</div><div><div><br /></div><div>// Create an empty ArrayList of type Integer</div><div>ArrayList&lt;Integer&gt; numbers = new ArrayList&lt;&gt;();</div><div><br /></div><div>// Create an ArrayList with initial values</div><div>ArrayList&lt;String&gt; fruits = new ArrayList&lt;&gt;(Arrays.asList("Apple", "Banana", "Orange"));</div></div><div><br /></div><div>3. Adding Elements:&nbsp;</div><div><br /></div><div>The add method can be used to add elements to an ArrayList.</div><div><div><br /></div><div>numbers.add(5);</div><div>numbers.add(10);</div><div>numbers.add(15);</div></div><div><br /></div><div>4. Accessing Elements:&nbsp;</div><div><br /></div><div>The get method allows you to access elements by their index.</div><div><div><br /></div><div>String fruit = fruits.get(0);&nbsp; // Gets the first element ("Apple")</div></div><div><br /></div><div>5. Removing Elements:</div><div><br /></div><div>You can eliminate elements by object or index using the remove method.</div><div><br /></div><div><div>fruits.remove(1);&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;// Removes the element at index 1 ("Banana")</div><div>fruits.remove("Orange");&nbsp; &nbsp; &nbsp; // Removes the element with the value "Orange"</div></div><div><br /></div><div>6. Iterating through Elements:&nbsp;</div><div><br /></div><div>There are several ways to iterate through an ArrayList's elements.</div><div><br /></div><div><div>for (Integer num : numbers) {</div><div>&nbsp; &nbsp; System.out.println(num);</div><div>}</div><div><br /></div><div>for (int i = 0; i &lt; fruits.size(); i++) {</div><div>&nbsp; &nbsp; System.out.println(fruits.get(i));</div><div>}</div></div><div><br /></div><div>7. Other Useful Methods:</div><div><br /></div><div>ArrayList has a number of methods for doing basic tasks including determining whether a list contains an element, locating an element's index, cleaning the list, and more.</div><div><br /></div><div><div>boolean containsApple = fruits.contains("Apple");</div><div>int indexBanana = fruits.indexOf("Banana");</div><div>fruits.clear();&nbsp; // Removes all elements from the ArrayList</div></div><div><br /></div><div><div>It's important to keep in mind that ArrayList automatically adjusts its size whenever new or removed elements are added, so you don't need to worry about doing it manually.</div><div><br /></div><div>The kind of components the ArrayList will contain (ArrayListInteger&gt; or ArrayListString&gt; in the instances above) is specified here using Java Generics. This guarantees type safety throughout compilation.</div></div><div><br /></div><div>8. Capacity and Performance:&nbsp;</div><div><br /></div><div>As new or discarded elements are added, the ArrayList automatically adjusts its size. By default, the starting capacity is set to 10, but you can change it if you expect to add a different number of pieces. Since resizing necessitates the creation of a new, larger array and the copying of elements, initializing the ArrayList with the appropriate capacity can enhance efficiency if you know the size in advance.</div><div><br /></div><div><div>ArrayList&lt;Integer&gt; numbers = new ArrayList&lt;&gt;(20);&nbsp; // Initial capacity of 20</div></div><div><br /></div><div>9. Iterating with Iterator:&nbsp;</div><div><br /></div><div>You can iterate through the elements in an ArrayList by using the Iterator interface. This gives you more control over the iteration and can be helpful if you want to remove elements as you go along.</div><div><br /></div><div><div>Iterator&lt;String&gt; iterator = fruits.iterator();</div><div>while (iterator.hasNext()) {</div><div>&nbsp; &nbsp; String fruit = iterator.next();</div><div>&nbsp; &nbsp; System.out.println(fruit);</div><div>}</div></div><div><br /></div><div>10. Bulk Operations:</div><div><br /></div><div>ArrayList allows for bulk actions such as adding or removing a number of entries at once.</div><div><div><br /></div><div>ArrayList&lt;String&gt; newFruits = new ArrayList&lt;&gt;(Arrays.asList("Grapes", "Watermelon"));</div><div>fruits.addAll(newFruits);&nbsp; // Adds all elements from newFruits to fruits</div><div><br /></div><div>fruits.removeAll(newFruits);&nbsp; // Removes all elements present in newFruits from fruits</div></div><div><br /></div><div>11. Sublist:&nbsp;</div><div><br /></div><div>The subList method can be used to extract a sublist from an ArrayList.</div><div><div><br /></div><div>List&lt;String&gt; sublist = fruits.subList(1, 3);&nbsp; // Gets elements at index 1 and 2</div></div><div><br /></div><div>12. Sorting:&nbsp;</div><div><br /></div><div>The Collections.sort method can be used to order the components of an ArrayList.</div><div><br /></div><div><div>Collections.sort(numbers);&nbsp; // Sorts the list of numbers in ascending order</div></div><div><br /></div><div>13. Conversion:&nbsp;</div><div><br /></div><div>Using the toArray method, you can change an ArrayList into a standard array.</div><div><br /></div><div><div>String[] fruitArray = fruits.toArray(new String[0]);</div></div><div><br /></div><div><div>14. Performance consideration:&nbsp;</div><div><br /></div><div>While ArrayList offers quick insertion/removal at the end and efficient random access in the middle, it is slower than other data structures like LinkedList. Consider utilizing a LinkedList if you require frequent additions and deletions at different places.</div><div><br /></div><div>15. Thread Safety:</div><div><br /></div><div>ArrayList is not thread-safe by default because it is not synchronized. Consider creating a synchronized version of an ArrayList using Collections.synchronizedList if you must use it in a multithreaded environment.</div></div><div><br /></div><div><div>List&lt;String&gt; synchronizedList = Collections.synchronizedList(new ArrayList&lt;&gt;());</div></div><div><br /></div><div>Just one of the numerous data structures offered by Java's Collections Framework, each with unique advantages and applications, is the ArrayList. Based on the particular program requirements, select the data format that best meets your needs.</div></description><link>https://www.knowitanything.com/2023/08/arraylist-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-1318463467975977245</guid><pubDate>Mon, 14 Aug 2023 05:21:00 +0000</pubDate><atom:updated>2023-08-14T10:51:39.686+05:30</atom:updated><title>Basics of Java</title><description><p>The world of Java programming is yours to explore. We'll discuss the essential Java building elements that each program is composed of in this chapter. Writing efficient and useful code requires an awareness of these fundamentals, regardless of your level of programming experience or familiarity with Java.</p><p>Syntax and Structure of Java Programs:</p><p>Java applications are based on a structured syntax that aids in logically organizing code. We'll go into the essential elements of a Java program in this section and investigate how they combine to produce executable apps.</p><p></p><ul style="text-align: left;"><li>Naming conventions and file organization for Java source files</li><li>Package Declarations: Creating packages from code</li><li>Class Definitions: Adding classes and methods</li><li>The major Technique: program entry point for Java</li></ul><p></p><div>Constants, Variables, and Data Types:</div><div><div><br /></div><div>The types of values that a variable can hold are determined by the different data types that Java allows. Understanding these data types is essential for correct data representation and optimal memory consumption.</div></div><div>Primitive data types:&nbsp;byte, short, int, long, float, double, char, boolean are examples of primitive data types.</div><div>Declaring variables: Naming conventions and initializing variables when declaring variables</div><div>Type casting: both tacit and overt casting</div><div>Constants: Declaring constants using the final keyword</div><div><br /></div><div><div>Expressions and Operators:</div><div><br /></div><div>You can execute a variety of operations on variables and values using operators. Expressions combine elements like operators and variables to produce meaningful results.</div></div><div><div><br /></div><div>Arithmetic Operators: +, -, *, /, %</div><div>Relational Operators: ==, !=, &lt;, &gt;, &lt;=, &gt;=</div><div>Logical Operators: &amp;&amp;, ||, !</div><div>Assignment Operators: =, +=, -=, *=, /=</div><div>Increment and Decrement Operators: ++, --</div><div>Conditional (Ternary) Operator: ? :</div><div>Bitwise Operators: &amp;, |, ^, ~, &lt;&lt;, &gt;&gt;</div></div><div><br /></div><div>Control Flow: Conditionals and Loops</div><div><div>You can specify the sequence in which statements are performed using control flow structures, which enables your software to make decisions and repeat actions.</div></div><div><div><br /></div><div>if Statement: Making decisions based on conditions</div><div>else and else if: Handling multiple conditions</div><div>switch Statement: Multi-branch decision-making</div><div>while Loop: Repeating code while a condition is true</div><div>do-while Loop: Executing code at least once</div><div>for Loop: Iterating a specific number of times</div><div>break and continue: Altering loop behavior</div></div><div><br /></div><div><div>You'll develop a strong understanding of the fundamental components that make up Java applications as you read through this chapter. You'll be prepared to move on to more complicated subjects in the subsequent chapters if you master these fundamentals. To reinforce your understanding, explore with coding examples and exercises because practice makes perfect.</div></div><div><br /></div></description><link>https://www.knowitanything.com/2023/08/basics-of-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-6722113921890324737</guid><pubDate>Mon, 14 Aug 2023 03:43:00 +0000</pubDate><atom:updated>2023-08-14T09:13:30.856+05:30</atom:updated><title>Constructors in Java</title><description><p>In Java, constructors are specialized methods used to initialize objects within a class. They offer a way to specify initial values for the object's attributes and are called whenever an object of the class is created. Constructors do not have a return type, not even void, and have the same name as the class. Here is how Java constructors operate:</p><p>public class MyClass {</p><p>&nbsp; &nbsp; private int value;</p><p>&nbsp; &nbsp; // Default Constructor</p><p>&nbsp; &nbsp; public MyClass() {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; value = 0;</p><p>&nbsp; &nbsp; }</p><p>&nbsp; &nbsp; // Parameterized Constructor</p><p>&nbsp; &nbsp; public MyClass(int initialValue) {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; value = initialValue;</p><p>&nbsp; &nbsp; }</p><p>&nbsp; &nbsp; // Methods and other members of the class...</p><p>&nbsp; &nbsp; public int getValue() {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; return value;</p><p>&nbsp; &nbsp; }</p><p>&nbsp; &nbsp; public void setValue(int newValue) {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; value = newValue;</p><p>&nbsp; &nbsp; }</p><p>}</p><p><br /></p><p>The class MyClass in the preceding example has two constructors:</p><p>Default Constructor: When an object is created without any arguments being passed, the default constructor is used, which has no parameters. The value attribute here has a starting value of 0.</p><p>Parameterized Constructor: This constructor initializes the value attribute with the supplied value and accepts an integer parameter.</p><p>These constructors can be used to produce class objects:</p><p>MyClass obj1 = new MyClass(); // Using the default constructor</p><p>MyClass obj2 = new MyClass(10); // Using the parameterized constructor</p><div><div>Among other things, constructors can be used to establish connections, assign resources, perform setup actions, and set initial values. Java will automatically create a default constructor (with no parameters and limited functionality) if you don't explicitly specify one in your class. However, the default constructor won't be inserted automatically if you declare any other ones.</div><div><br /></div><div>Constructor chaining is another option, where one constructor uses the this() keyword to invoke another. You can reuse code in this way and speed up object initialization.</div></div><div><div><br /></div><div>public class MyClass {</div><div>&nbsp; &nbsp; private int value;</div><div><br /></div><div>&nbsp; &nbsp; public MyClass() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; this(0); // Calls the parameterized constructor with default value</div><div>&nbsp; &nbsp; }</div><div><br /></div><div>&nbsp; &nbsp; public MyClass(int initialValue) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; value = initialValue;</div><div>&nbsp; &nbsp; }</div><div><br /></div><div>&nbsp; &nbsp; // Other methods...</div><div>}</div></div><div><br /></div><div>In your Java programs, keep in mind that constructors help create properly initialized and consistent objects.</div></description><link>https://www.knowitanything.com/2023/08/constructors-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7735280183003054529</guid><pubDate>Mon, 14 Aug 2023 03:33:00 +0000</pubDate><atom:updated>2023-08-14T09:03:42.028+05:30</atom:updated><title>Exception handling in java</title><description><p>A feature called exception handling in Java enables you to handle and control errors or extraordinary circumstances that could happen while a program is being executed. Objects that represent exceptional circumstances that may impede the regular course of code execution are known as exceptions. Java has an integrated exception handling system that can deal with these circumstances and make sure the program doesn't crash abruptly.</p><p>An overview of Java's exception handling process is provided below:</p><p>Types of Exceptions:</p><p>According to Java, there are two basic exception types: checked exceptions and unchecked exceptions (sometimes referred to as runtime exceptions).</p><p>Checked Exceptions: These are errors that the compiler expects you to explicitly catch with try-catch blocks or by using the throws keyword to indicate that your method throws the error.</p><p>Unchecked Exceptions: They (also known as runtime exceptions) do not need to be handled manually, but try-catch blocks can still be used to catch them.</p><p>Try-catch Blocks:</p><p>Code that could raise an exception is contained in the try block of a try-catch statement. If an exception arises, the code to handle it is located in the catch block. To handle various error kinds, you can have numerous catch blocks.</p><p>try {</p><p>&nbsp; &nbsp; // Code that might throw an exception</p><p>} catch (ExceptionType1 e1) {</p><p>&nbsp; &nbsp; // Handle ExceptionType1</p><p>} catch (ExceptionType2 e2) {</p><p>&nbsp; &nbsp; // Handle ExceptionType2</p><p>} finally {</p><p>&nbsp; &nbsp; // Optional: Code that runs regardless of whether an exception occurred or not</p><p>}</p><div>Throw Statement:&nbsp;</div><div>You can manually throw an exception from your code using the throw statement. When you wish to show that a particular error situation has occurred, this is helpful.</div><div><div><br /></div><div>if (condition) {</div><div>&nbsp; &nbsp; throw new CustomException("This is a custom exception");</div><div>}</div></div><div><br /></div><div>throws Keyword:&nbsp;</div><div>In the method signature, a method must be declared if it has the ability to throw an exception. Callers are informed of the mechanism they must use to handle or propagate the exception through this.</div><div><div><br /></div><div>public void someMethod() throws SomeException {</div><div>&nbsp; &nbsp; // Method code that might throw SomeException</div><div>}</div></div><div><div><br /></div><div>finally Block:&nbsp;</div><div>Regardless of whether an exception was thrown or not, code that must be run is specified in the finally block. It is frequently applied to cleanup tasks like shutting off resources.</div><div><br /></div><div>Custom Exceptions:&nbsp;</div><div>By extending the default Exception class or one of its subclasses, you can create your own custom exception classes. This is helpful for developing particular exception kinds that are related to the domain of your application.</div></div><div><div><br /></div><div>public class CustomException extends Exception {</div><div>&nbsp; &nbsp; // Constructor(s) and additional methods</div><div>}</div></div><div><br /></div><div>Java's exception handling features let you gracefully handle failures, increase application robustness, and give users helpful error messages.</div><div><br /></div></description><link>https://www.knowitanything.com/2023/08/a-feature-called-exception-handling-in.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-24762278820727924</guid><pubDate>Mon, 14 Aug 2023 03:19:00 +0000</pubDate><atom:updated>2023-08-14T08:49:56.393+05:30</atom:updated><title>Multithreading in Java</title><description><p>&nbsp;The Java programming language's capacity to run several threads concurrently within a single process is referred to as multithreading. The lowest unit of execution within a process is a thread. Applications that need the execution of multiple tasks concurrently or in parallel will perform better thanks to multithreading, which makes greater use of the CPU resources at their disposal.</p><p>The fundamental ideas and elements of Java's multithreading are as follows:</p><p>Thread Class and Runnable Interface:&nbsp;</p><p>By either extending the Thread class or realizing the Runnable interface, you can construct threads in Java. It is frequently recommended to implement the Runnable interface since it enables greater concern separation and gets around Java's single inheritance restriction.</p><p>Thread Lifecycle:</p><p>Throughout their lifespan, threads can be in several states, including NEW, RUNNABLE, BLOCKED, WAITING, TIMED_WAITING, and TERMINATED. To manage and control thread states, you can employ a variety of techniques.</p><p>Creating and Starting Threads:</p><p>You must extend the Thread class and override the run() function in order to create and start a thread using it. Calling the start() method on the thread class instance starts the thread. You can also make a Runnable object and provide it to a thread instance as an alternative.</p><p>Thread Synchronization:</p><p>Synchronization is required when multiple threads access shared resources simultaneously in order to prevent race situations and ensure data integrity. In addition to more sophisticated synchronization utilities offered by the java.util.concurrent package, Java also includes techniques like synchronized methods and blocks.</p><p>Thread Priorities:</p><p>The setPriority() method allows for the assignment of priorities to threads. Because thread execution order is platform-dependent, thread priorities might not always provide perfect control.</p><p>Thread Communication:</p><p>Through the use of functions like wait(), notify(), and notifyAll(), threads can communicate with one another. These techniques enable threads to delay execution until a number of criteria have been met.</p><p>Thread Pooling:</p><p>Due to the overhead of thread creation and administration, starting a new thread for each job may not be efficient. You can submit tasks for execution using Java's Executor framework, which controls a pool of reusable threads.</p><p>Daemon Threads:</p><p>Daemon threads are background-running threads that don't obstruct a program's ability to terminate. They frequently carry out duties like rubbish pickup.</p><p>Thread Safety:</p><p>Designing your code and data structures to avoid race situations and other synchronization-related problems is one method to ensure thread safety. In order to preserve thread safety, proper synchronization techniques are essential.</p><p>Applications that use tasks that can be parallelized can perform much better thanks to Java's multithreading capabilities. However, it also adds complications for thread synchronization and coordination. For the purpose of creating multithreaded Java applications that are reliable and effective, it is crucial to fully comprehend these ideas.</p></description><link>https://www.knowitanything.com/2023/08/multithreading-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-1358678748424399094</guid><pubDate>Sun, 13 Aug 2023 08:45:00 +0000</pubDate><atom:updated>2023-08-13T14:15:43.871+05:30</atom:updated><title>Abstraction and Interfaces in Java</title><description><p>&nbsp;Abstraction and interfaces are two crucial Java concepts that support the creation of modular and adaptable program designs. Let's investigate each of these ideas:</p><p>Abstraction:&nbsp;</p><p>A key idea of object-oriented programming is abstraction, which focuses on concealing intricate implementation details and exposing only the necessary characteristics of an object. You can use it to produce a condensed illustration of an object's activity. Through abstract classes and methods, this is accomplished.</p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="background-color: white; white-space-collapse: preserve;">Abstract classes: </span><span style="white-space-collapse: preserve;">A class that cannot be instantiated on its own is said to be abstract. It acts as a template from which other classes can inherit. Both abstract methods (methods without a body) and concrete methods (methods with implementation) can be found in abstract classes. All the abstract methods declared in an abstract class must have implementations in subclasses that derive from it.</span></span></p><p>Example:</p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">abstract class Shape {</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; abstract double area(); // Abstract method without implementation</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; void display() {</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; &nbsp; &nbsp; System.out.println("Displaying shape.");</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; }</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">}</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">class Circle extends Shape {</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; double radius;</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; Circle(double radius) {</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; &nbsp; &nbsp; this.radius = radius;</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; }</span></span><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, &quot;Segoe UI&quot;, Roboto, Ubuntu, Cantarell, &quot;Noto Sans&quot;, sans-serif, &quot;Helvetica Neue&quot;, Arial, &quot;Apple Color Emoji&quot;, &quot;Segoe UI Emoji&quot;, &quot;Segoe UI Symbol&quot;, &quot;Noto Color Emoji&quot;; white-space-collapse: preserve;">&nbsp;&nbsp;</span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; @Override</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; double area() {</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; &nbsp; &nbsp; return Math.PI * radius * radius;</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">&nbsp; &nbsp; }</span></span></p><p></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">}</span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">Interfaces: </span></span></p><p><span style="font-family: Söhne, ui-sans-serif, system-ui, -apple-system, Segoe UI, Roboto, Ubuntu, Cantarell, Noto Sans, sans-serif, Helvetica Neue, Arial, Apple Color Emoji, Segoe UI Emoji, Segoe UI Symbol, Noto Color Emoji;"><span style="white-space-collapse: preserve;">An interface is a set of methods that a class is required to implement. Because a class can implement multiple interfaces, it enables multiple inheritance in Java. Polymorphism and loose coupling are made possible by the way interfaces ensure that classes follow a specific contract.</span></span></p><div>Interface Declaration:</div><div><div><br /></div><div>interface Drawable {</div><div>&nbsp; &nbsp; void draw(); // Method signatures without implementation</div><div>&nbsp; &nbsp; void resize(int factor);</div><div>}</div></div><div><br /></div><div>Class Implementing an Interface:</div><div><div>class Circle implements Drawable {</div><div>&nbsp; &nbsp; private double radius;</div><div>&nbsp; &nbsp;&nbsp;</div><div>&nbsp; &nbsp; Circle(double radius) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; this.radius = radius;</div><div>&nbsp; &nbsp; }</div><div>&nbsp; &nbsp;&nbsp;</div><div>&nbsp; &nbsp; @Override</div><div>&nbsp; &nbsp; public void draw() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; System.out.println("Drawing circle.");</div><div>&nbsp; &nbsp; }</div><div>&nbsp; &nbsp;&nbsp;</div><div>&nbsp; &nbsp; @Override</div><div>&nbsp; &nbsp; public void resize(int factor) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; radius *= factor;</div><div>&nbsp; &nbsp; }</div><div>}</div></div><div><br /></div><div>Interface Inheritance:</div><div>You can create a hierarchy of interfaces since interfaces can extend one another.</div><div><div><br /></div><div>interface Shape extends Drawable {</div><div>&nbsp; &nbsp; double area();</div><div>}</div></div><div><br /></div><div>In conclusion, abstraction enables you to set contracts that classes must abide by, increasing flexibility and modularity in your code, whereas interfaces give you a mechanism to define such contracts. When creating extendable and maintainable Java applications, both ideas are essential.</div></description><link>https://www.knowitanything.com/2023/08/abstraction-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-7910529617460581644</guid><pubDate>Sun, 13 Aug 2023 08:34:00 +0000</pubDate><atom:updated>2023-08-13T14:04:33.795+05:30</atom:updated><title>Encapsulation in Java</title><description><p>One of the guiding principles of object-oriented programming (OOP) is encapsulation, which is mostly related to programming languages like Java. It refers to the process of grouping together into a single entity called a class data (attributes) and the methods (functions) that operate on that data. When constructing objects, which are instances of the class, the class serves as a blueprint.</p><p>The advantages of encapsulation are as follows:</p><p>Data hiding: You can manage the visibility and accessibility of the data by enclosing it within a class. You can select which characteristics are private (only accessible within the class) and which are public (visible from outside the class). This aids in preventing unauthorized or inadvertent changes to an object's internal state.</p><p>Abstraction: By encapsulating an item, you can reveal a straightforward user interface while concealing its intricate internal implementation details. The code is made more maintainable as a result of the promotion of a distinct separation between the interface and the implementation.</p><p>Code Organisation:&nbsp;Coding structure and organization are improved by combining data and related methods into a single class. The codebase is now simpler to comprehend, adapt, and expand as a result.</p><p>Encapsulation in Java is accomplished using getter/setter methods and access modifiers:</p><div><div>Private Access Modifier: You can designate attributes as private to limit their accessibility to the class by using the Private Access Modifier. This keeps anybody outside of the class from having direct access to the attributes.</div><div><br /></div><div>Public Access Modifier: To grant regulated access to the private attributes, you can construct public methods (getter and setter methods). An attribute's value can be retrieved using getter methods, and it can be changed using setter methods.</div></div><div><br /></div><div>Here is a simple Java example that illustrates encapsulation:</div><div><br /></div><div><div>public class Student {</div><div>&nbsp; &nbsp; private String name;</div><div>&nbsp; &nbsp; private int age;</div><div><br /></div><div>&nbsp; &nbsp; public String getName() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; return name;</div><div>&nbsp; &nbsp; }</div><div><br /></div><div>&nbsp; &nbsp; public void setName(String newName) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; name = newName;</div><div>&nbsp; &nbsp; }</div><div><br /></div><div>&nbsp; &nbsp; public int getAge() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; return age;</div><div>&nbsp; &nbsp; }</div><div><br /></div><div>&nbsp; &nbsp; public void setAge(int newAge) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; if (newAge &gt;= 0) {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; age = newAge;</div><div>&nbsp; &nbsp; &nbsp; &nbsp; }</div><div>&nbsp; &nbsp; }</div><div>}</div></div><div><br /></div><div><div>The Student class in this example contains the name and age attributes. Since they are private, nobody outside of the class can directly access them. Instead, these characteristics are interacted with via the getter and setter methods. To make sure the age is not negative, the age setter performs validation.</div><div><br /></div><div>Encapsulation offers a regulated manner to interact with an object's data and aids in protecting the integrity of the internal state of the class.</div></div><div><br /></div></description><link>https://www.knowitanything.com/2023/08/encapsulation-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item><item><guid isPermaLink="false">tag:blogger.com,1999:blog-6615370946101387199.post-4310541070419416250</guid><pubDate>Sun, 13 Aug 2023 08:27:00 +0000</pubDate><atom:updated>2023-08-13T13:57:49.020+05:30</atom:updated><title>Polymorphism in Java</title><description><p>A fundamental idea in object-oriented programming, polymorphism is a vital component of the Java programming language. It enables more flexible and general code design by allowing objects of several classes to be viewed as belonging to a single superclass.</p><p>Java supports runtime (or dynamic) polymorphism in addition to compile-time (or static) polymorphism.</p><p>Compile-Time Polymorphism (Method Overloading):&nbsp;</p><p>Method overloading, also known as compile-time polymorphism, happens when many methods in the same class have the same name but different parameters. Based on the quantity and kind of arguments given during the method call, the Java compiler chooses which method to invoke.</p><p>class Calculator {</p><p>&nbsp; &nbsp; int add(int a, int b) {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; return a + b;</p><p>&nbsp; &nbsp; }&nbsp; &nbsp;</p><p>&nbsp; &nbsp; double add(double a, double b) {</p><p>&nbsp; &nbsp; &nbsp; &nbsp; return a + b;</p><p>&nbsp; &nbsp; }</p><p>}</p><div><div>The add method in this illustration is overloaded to accept both integer and double parameters.</div><div><br /></div><div>Runtime Polymorphism (Method Overriding):</div><div><br /></div><div>When a subclass offers a particular implementation for a method that is already specified in its superclass, this phenomenon is known as runtime polymorphism (method overriding). Based on the actual object type, a choice is made about which method to invoke at runtime.</div></div><div><div><br /></div><div>class Shape {</div><div>&nbsp; &nbsp; void draw() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; System.out.println("Drawing a shape");</div><div>&nbsp; &nbsp; }</div><div>}</div><div><br /></div><div>class Circle extends Shape {</div><div>&nbsp; &nbsp; @Override</div><div>&nbsp; &nbsp; void draw() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; System.out.println("Drawing a circle");</div><div>&nbsp; &nbsp; }</div><div>}</div><div><br /></div><div>class Square extends Shape {</div><div>&nbsp; &nbsp; @Override</div><div>&nbsp; &nbsp; void draw() {</div><div>&nbsp; &nbsp; &nbsp; &nbsp; System.out.println("Drawing a square");</div><div>&nbsp; &nbsp; }</div><div>}</div></div><div><div><br /></div><div>In this illustration, the draw method from the Shape class is overridden by the Circle and Square classes. Depending on the kind of object being referenced to at runtime, a specific method must be called.</div><div><br /></div><div>In Java, polymorphism is accomplished through interfaces and method overriding. To remember, have the following in mind:</div></div><div><div><ul style="text-align: left;"><li>In order to achieve code reuse and develop adaptable, extendable programs, polymorphism is a necessity.</li><li>The method declaration is located in the base class (superclass), while the method implementations are provided by the subclasses (method overriding).</li><li>A method in a subclass that is meant to override a method in the superclass is indicated by the @Override annotation. If there is a mismatch in the method signatures, this aids in the compilation of errors.</li><li>Because of polymorphism, you can create code that interacts with objects of various subclasses without having to understand their precise types.</li></ul><div>In conclusion, Java's polymorphism feature enables you to write more modular, adaptable, and manageable code by allowing various objects to be handled uniformly regardless of their true types.</div></div></div></description><link>https://www.knowitanything.com/2023/08/polymorphism-in-java.html</link><author>noreply@blogger.com (study engineering)</author><thr:total>0</thr:total></item></channel></rss>