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Book 6 in the Princeton Mathematical Series. Originally published in 1941. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material.
The purpose of this book is to give an introduction to the Laplace transform on the undergraduate level. The material is drawn from notes for a course taught by the author at the Milwaukee School of Engineering. Based on classroom experience, an attempt has been made to (1) keep the proofs short, (2) introduce applications as soon as possible, (3) concentrate on problems that are difficult to handle by the older classical methods, and (4) emphasize periodic phenomena. To make it possible to offer the course early in the curriculum (after differential equations), no knowledge of complex variable theory is assumed. However, since a thorough study of Laplace. transforms requires at least the rudiments of this theory, Chapter 3 includes a brief sketch of complex variables, with many of the details presented in Appendix A. This plan permits an introduction of the complex inversion formula, followed by additional applications. The author has found that a course taught three hours a week for a quarter can be based on the material in Chapters 1, 2, and 5 and the first three sections of Chapter 7. If additional time is available (e.g., four quarter-hours or three semester-hours), the whole book can be covered easily. The author is indebted to the students at the Milwaukee School of Engineering for their many helpful comments and criticisms.
The aim of this comparatively short textbook is a sufficiently full exposition of the fundamentals of the theory of functions of a complex variable to prepare the student for various applications. Several important applications in physics and engineering are considered in the book. This thorough presentation includes all theorems (with a few exceptions) presented with proofs. No previous exposure to complex numbers is assumed. The textbook can be used in one-semester or two-semester courses. In one respect this book is larger than usual, namely in the number of detailed solutions of typical problems. This, together with various problems, makes the book useful both for self- study and for the instructor as well. A specific point of the book is the inclusion of the Laplace transform. These two topics are closely related. Concepts in complex analysis are needed to formulate and prove basic theorems in Laplace transforms, such as the inverse Laplace transform formula. Methods of complex analysis provide solutions for problems involving Laplace transforms. Complex numbers lend clarity and completion to some areas of classical analysis. These numbers found important applications not only in the mathematical theory, but in the mathematical descriptions of processes in physics and engineering.
Clear explanations and supportive online material develop an intuitive understanding of the meaning and use of Laplace.
Acclaimed text on engineering math for graduate students covers theory of complex variables, Cauchy-Riemann equations, Fourier and Laplace transform theory, Z-transform, and much more. Many excellent problems.
The classical theory of the Laplace Transform can open many new avenues when viewed from a modern, semi-classical point of view. In this book, the author re-examines the Laplace Transform and presents a study of many of the applications to differential equations, differential-difference equations and the renewal equation.
Classic graduate-level exposition covers theory and applications to ordinary and partial differential equations. Includes derivation of Laplace transforms of various functions, Laplace transform for a finite interval, and more. 1948 edition.
This book gives background material on the theory of Laplace transforms, together with a fairly comprehensive list of methods that are available at the current time. Computer programs are included for those methods that perform consistently well on a wide range of Laplace transforms. Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations.