The Laplace Transform

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 accessi ble to undergraduates. To this end the text addresses a number of issues which are rarely considered. Special features of the book include a justification of the Laplace Transform method for solving linear ODEs, sufficient conditions for taking the Laplace Transform of an infinite series, steady-state solutions, and a rigorous approach to the Complex Inversion Formula, to name but a few.

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.

Résumé
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.

Contenu
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.