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4300 results in General and Classical Physics

Page 21 of 215

  • 2 - Laplace-Transform Examples

  • Daniel Fleisch, Wittenberg University, Ohio
  • Book:
    A Student's Guide to Laplace Transforms
    Published online:
    11 January 2022
    Print publication:
    13 January 2022, pp 40-91

  • Summary

    Becoming familiar with the Laplace transform F(s) of basic time-domain functions f(t) such as exponentials, sinusoids, powers of t, and hyperbolic functions can be immensely useful in a variety of applications. That is because many of the more complicated functions that describe the behavior of real-world systems and that appear in differential equations can be synthesized as a mixture of these basic functions. And although there are dozens of books and websites that show you how to find the Laplace transform of such functions, much harder to find are explanations that help you achieve an intuitive understanding of why F(s) takes the form it does, that is, an understanding that goes beyond “That’s what the integral gives”. So the goal of this chapter is not just to show you the Laplace transforms of some basic functions, but to provide explanations that will help you see why those transforms make sense.

  • 1 - The Fourier and Laplace Transforms

  • Daniel Fleisch, Wittenberg University, Ohio
  • Book:
    A Student's Guide to Laplace Transforms
    Published online:
    11 January 2022
    Print publication:
    13 January 2022, pp 1-39

  • Summary

    The Laplace transform is a mathematical operation that converts a function from one domain to another. And why would you want to do that? As you’ll see in this chapter, changing domains can be immensely helpful in extracting information from the mathematical functions and equations that describe the behavior of natural phenomena as well as mechanical and electrical systems. Specifically, when the Laplace transform operates on a function f(t) that depends on the parameter t, the result of the operation is a function F(s) that depends on the parameter s. You’ll learn the meaning of those parameters as well as the details of the mathematical operation that is defined as the Laplace transform in this chapter, and you’ll see why the Fourier transform can be considered to be a special case of the Laplace transform.

  • 3 - Properties of the Laplace Transform

  • Daniel Fleisch, Wittenberg University, Ohio
  • Book:
    A Student's Guide to Laplace Transforms
    Published online:
    11 January 2022
    Print publication:
    13 January 2022, pp 92-120

  • Summary

    The value of knowing the Laplace transforms of the basic functions described in the previous chapter is greatly enhanced by certain properties of the Laplace transform. That is because these properties allow you to determine the transform of much more complicated time-domain functions by combining and modifying the transforms of simple functions such as those discussed in .

A Student's Guide to Laplace Transforms
  • Daniel Fleisch
  • Published online:
    11 January 2022
    Print publication:
    13 January 2022
  • The Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student's Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section of the book.

A Student's Guide to the Schrödinger Equation
  • Daniel A. Fleisch
  • Published online:
    08 September 2021
    Print publication:
    20 February 2020
  • Quantum mechanics is a hugely important topic in science and engineering, but many students struggle to understand the abstract mathematical techniques used to solve the Schrödinger equation and to analyze the resulting wave functions. Retaining the popular approach used in Fleisch's other Student's Guides, this friendly resource uses plain language to provide detailed explanations of the fundamental concepts and mathematical techniques underlying the Schrödinger equation in quantum mechanics. It addresses in a clear and intuitive way the problems students find most troublesome. Each chapter includes several homework problems with fully worked solutions. A companion website hosts additional resources, including a helpful glossary, Matlab code for creating key simulations, revision quizzes and a series of videos in which the author explains the most important concepts from each section of the book.

Page 21 of 215