Introduction to the Laplace Transform

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Springer Science & Business Media, Apr 1, 1978 - Mathematics - 205 pages
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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
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