Synopses & Reviews
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.
Review
"This remarkable textbook for students of either mathematics or engineering deals with a rigorous, but well readable, treatment of the basic properties of the Laplace transform and its applications to various types of equations up to partial differential equations, in particular to those stemming from physics and engineering. The necessary mathematical remedies are explained in detail, e.g. the elements of the theory of analytical functions and those of the Riemann-Stieltjes integral in order to have a rigorous basis for the treatment of the complex inversion formula and the Dirac delta function, respectively. The book contains several exercises with solutions and a table of the usual Laplace transforms. ZENTRALBLATT MATH"
Review
"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
Synopsis
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.
Description
Includes bibliographical references (p. 207-208) and index.
Table of Contents
Basic Principles.- Applications and Properties.- Complex Variable Theory.- Complex Inversion Formula.- Partial Differential Equations.- Appendix.- References.