# Introduction to the Laplace Transform

Springer Science & Business Media, Apr 1, 1978 - Mathematics - 205 pages
The purpose of this book is to give an introduction to the Laplace transform on the undergraduate level. The material is drawn from notes for a course taught by the author at the Milwaukee School of Engineering. Based on classroom experience, an attempt has been made to (1) keep the proofs short, (2) introduce applications as soon as possible, (3) concentrate on problems that are difficult to handle by the older classical methods, and (4) emphasize periodic phenomena. To make it possible to offer the course early in the curriculum (after differential equations), no knowledge of complex variable theory is assumed. However, since a thorough study of Laplace. transforms requires at least the rudiments of this theory, Chapter 3 includes a brief sketch of complex variables, with many of the details presented in Appendix A. This plan permits an introduction of the complex inversion formula, followed by additional applications. The author has found that a course taught three hours a week for a quarter can be based on the material in Chapters 1, 2, and 5 and the first three sections of Chapter 7. If additional time is available (e.g., four quarter-hours or three semester-hours), the whole book can be covered easily. The author is indebted to the students at the Milwaukee School of Engineering for their many helpful comments and criticisms.

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### Contents

 Basic Properties and Applications 1 12 The Variable s 3 13 Laplace Transforms of Some Special Functions 4 14 Some Basic Properties of the Laplace Transform 10 15 Inverse Laplace Transforms 18 16 Partial Fractions 21 17 Differential Equations 30 18 Applications 33
 32 The Residue Theorem 100 The Complex Inversion Formula 105 42 The Inversion Integral 107 Convolutions 111 52 Two Special Limits 118 53 Applications 119 Transforms with Infinitely Many Singularities 129 Applications to Partial Differential Equations 145

 19 Differentiation and Integration of Transforms 38 Further Properties and Applications 43 22 The Second Translation Theorem 45 23 Transforms by Graphical Addition 50 24 The Unit Impulse Function 55 25 Applications 63 26 Transforms of Periodic Functions 74 27 Applications 83 Sketch of Complex Variable Theory 93
 72 The Diffusion Equation 152 73 The Vibrating String 159 74 The Complex Inversion Formula Again 164 More on Complex Variable Theory 169 Table of Laplace Transforms 179 Bibliography 185 Answers to Exercises 187 Index 201 Copyright