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The Laplace Transform : Theory and Applications (Undergraduate Texts in Mathematics

by Joel L. Schiff
Publisher: Springer; 1 edition (October 14, 1999)
Language: English
ISBN: 0387986987
Book Description
The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial. It can also be used to solve difference equations. The present text, while mathematically rigorous, is readily accessible to students of either mathematics or engineering. Even the Dirac delta function, which is normally covered in a heuristic fashion, is given a completely justifiable treatment in the context of the Riemann-Stieltjes integral, yet at a level an undergraduate student can appreciate. When it comes to the deepest part of the theory, the Complex Inversion Formula, a knowledge of poles, residues, and contour integration of meromorphic functions is required. To this end, an entire chapter is devoted to the fundamentals of complex analysis. In addition to all the theoretical considerations, there are numerous worked examples drawn from engineering and physics.When applying the Laplace transform, it is important to have a good understanding of the theory underlying it, rather than just a cursory knowledge of its application. This text provides that understanding.
Book Info
Provides a mathematically rigorous, yet readily accessible treatment of applications of Laplace transforms. Includes the Dirac delta function given a completely justifiable treatment in the context of the Riemann-Stieltjes integral. DLC: Laplace transformation.
Contents
Preface ix
1 Basic Principles 1
1.1 The Laplace Transform . . . . . . . . . . . . . . . . . 1
1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Continuity Requirements . . . . . . . . . . . . . . . . 8
1.4 Exponential Order . . . . . . . . . . . . . . . . . . . . 12
1.5 The Class L . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Basic Properties of the Laplace Transform . . . . . . 16
1.7 Inverse of the Laplace Transform . . . . . . . . . . . 23
1.8 Translation Theorems . . . . . . . . . . . . . . . . . . 27
1.9 Differentiation and Integration of the
Laplace Transform . . . . . . . . . . . . . . . . . . . . 31
1.10 Partial Fractions . . . . . . . . . . . . . . . . . . . . . 35
2 Applications and Properties 41
2.1 Gamma Function . . . . . . . . . . . . . . . . . . . . 41
2.2 Periodic Functions . . . . . . . . . . . . . . . . . . . . 47
2.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Ordinary Differential Equations . . . . . . . . . . . . 59
2.5 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . 74
xiii
xiv Contents
2.6 Asymptotic Values . . . . . . . . . . . . . . . . . . . . 88
2.7 Convolution . . . . . . . . . . . . . . . . . . . . . . . . 91
2.8 Steady-State Solutions . . . . . . . . . . . . . . . . . . 103
2.9 Difference Equations . . . . . . . . . . . . . . . . . . 108
3 Complex Variable Theory 115
3.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . 115
3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . 136
3.5 Integrals of the Type ∞
−∞ f (x) dx . . . . . . . . . . . . 147
4 Complex Inversion Formula 151
5 Partial Differential Equations 175
Appendix 193
References 207
Tables 209
Laplace Transform Operations . . . . . . . . . . . . . . . . 209
Table of Laplace Transforms . . . . . . . . . . . . . . . . . . 210
Answers to Exercises 219
Index 231
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