# The Laplace Transform

## Theory and Applications

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xiv
2. Joel L. Schiff
Pages 1-39
3. Joel L. Schiff
Pages 41-114
4. Joel L. Schiff
Pages 115-150
5. Joel L. Schiff
Pages 151-174
6. Joel L. Schiff
Pages 175-191
7. Back Matter
Pages 193-235

### Introduction

The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor, and dubious mathematical practices are not uncommon in the literature for students. In the present text, I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to un­ dergraduates. Th this end, this text addresses a number of issues that are rarely considered. For instance, when we apply the Laplace trans­ form method to a linear ordinary differential equation with constant coefficients, any(n) + an-lY(n-l) + · · · + aoy = f(t), why is it justified to take the Laplace transform of both sides of the equation (Theorem A. 6)? Or, in many proofs it is required to take the limit inside an integral. This is always fraught with danger, especially with an improper integral, and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required. IX X Preface Furthermore, it is sometimes desirable to take the Laplace trans­ form of an infinite series term by term. Again it is shown that this cannot always be done, and specific sufficient conditions are established to justify this operation.

### Keywords

Dirac delta function complex analysis difference equation differential equation integral integration partial differential equation residue transform theory

#### Authors and affiliations

1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

### Bibliographic information

• Book Title The Laplace Transform
• Book Subtitle Theory and Applications
• Authors Joel L. Schiff
• Series Title Undergraduate Texts in Mathematics
• DOI https://doi.org/10.1007/978-0-387-22757-3
• Copyright Information Springer-Verlag New York 1999
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-98698-2
• Softcover ISBN 978-1-4757-7262-3
• eBook ISBN 978-0-387-22757-3
• Series ISSN 0172-6056
• Edition Number 1
• Number of Pages XIV, 236
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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