The Laplace Transform

Preliminary Text. Do not use. The Laplace transform is a wonderful tool for solving ordinary and partial differential equations, and has enjoyed much success in this realm. But with its success, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. In the present text, the author brings to the subject a certain amount of mathematical correctness and makes it accessible to undergraduates.

Klappentext

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.

Zusammenfassung
"The book contains plenty of examples, exercises (with answers at the end) and a table of transforms.
The book is certainly a handsome introduction to the Laplae transform, by its clear representation well-suited for self-study."
Nieuw Archief voor Wiskunde, March 2001

"This clearly written undergraduate textbook can be recommended to students and teachers of this subject, both in a mathematial and engineering context."
Newsletter of the EMS, issue 42, December 2001

Inhalt
1 Basic Principles.- 2 Applications and Properties.- 3 Complex Variable Theory.- 4 Complex Inversion Formula.- 5 Partial Differential Equations.- References.- Tables.- Laplace Transform Operations.- Table of Laplace Transforms.- Answers to Exercises.