Undergraduate Texts in Mathematics

Transcription

1 Undergraduate Texts in Mathematics Editors S. Axler F. W. Gehring K.A. Ribet Springer Science+Business Media, LLC

2 Undergraduate Texts in Mathematics Anglin: Mathematics: A Concise History and Philosophy. Readings in Mathematics. Anglin/Lambek: The Heritage of Thales. Readings in Mathematics. Apostol: Introduction to Analytic Number Theory. Second edition. Armstrong: Basic Topology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right. Second edition. Beardon: Limits: A New Approach to Real Analysis. Bak/Newman: Complex Analysis. Second edition. Banchoff!Wermer: Linear Algebra Through Geometry. Second edition. Berberian: A First Course in Real Analysis. Bix: Conics and Cubics: A Concretem Introduction to Algebraic Curves. Bremaud: An Introduction to Probabilistic Modeling. Bressoud: Factorization and Primality Testing. Bressoud: Second Year Calculus. Readings in Mathematics. Brickman: Mathematical Introduction to Linear Programming and Game Theory. Browder: Mathematical Analysis: An Introduction. Buskes/van Rooij: Topological Spaces: From Distance to Neighborhood. Cederberg: A Course in Modem Geometries. Childs: A Concrete Introduction to Higher Algebra. Second edition. Chung: Elementary Probability Theory with Stochastic Processes. Third edition. Cox/Little/O'Shea: Ideals, Varieties, and Algorithms. Second edition. Croom: Basic Concepts of Algebraic Topology. Curtis: Linear Algebra: An Introductory Approach. Fourth edition. Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory. Second edition. Dixmier: General Topology. Driver: Why Math? Ebbinghaus/Flumffhomas: Mathematical Logic. Second edition. Edgar: Measure, Topology, and Fractal Geometry. Elaydi: An Introduction to Difference Equations. Second edition. Exner: An Accompaniment to Higher Mathematics. Fine/Rosenberger: The Fundamental Theory of Algebra. Fischer: Intermediate Real Analysis. Flanigan/Kazdan: Calculus Two: Linear and Nonlinear Functions. Second edition. Fleming: Functions of Several Variables. Second edition. Foulds: Combinatorial Optimization for Undergraduates. Foulds: Optimization Techniques: An Introduction. Franklin: Methods of Mathematical Economics. Frazier: An Introduction to Wavelets Through Linear Algebra. Gordon: Discrete Probability. Hairer/Wanner: Analysis by Its History. Readings in Mathematics. Halmos: Finite-Dimensional Vector Spaces. Second edition. Halmos: Naive Set Theory. Hammerlin!Hoffmann: Numerical Mathematics. Readings in Mathematics. Hartshorne: Geometry: Euclid and Beyond. Hijab: Introduction to Calculus and Classical Analysis. Hilton!Holton/Pedersen: Mathematical Reflections: In a Room with Many Mirrors. Iooss/Joseph: Elementary Stability and Bifurcation Theory. Second edition. (continued after index)

3 Joel L. Schiff The Laplace Transform Theory and Applications With 68 Illustrations Springer

4 Joel L. Schiff Department of Mathematics The University of Auckland Private Bag Auckland New Zealand Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA USA F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI USA K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA USA Mathematics Subject Classification (1991 ): 44Al0 Library of Congress Cataloging-in-Publication Data Schiff, Joel L. The Laplace transform: theory and applications I Joel L. Schiff. p. cm.-(undergraduate texts in mathematics) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Laplace transformation. I. Title. II. Series. QA432.S dc Printed on acid-free paper Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcoverreprint ofthe bardeover Istedition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC. except for brief excerpts in connecfion with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe formerarenot especially identified, is nottobe taken as a sign that such names, as understood by the 'Itade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Robert Bruni; manufacturing supervised by Jeffrey Taub. 'I)rpeset by The Bartlett Press, Inc., Marietta, GA ISBN SPIN

5 1b my parents

6 It is customary to begin courses in mathematical engineering by explaining that the lecturer would never trust his life to an aeroplane whose behaviour depended on properties of the Lebesgue integral. It might, perhaps, be just as foolhardy to fly in an aeroplane designed by an engineer who believed that cookbook application of the Laplace transform revealed all that was to be known about its stability. T.W. Korner Fourier Analysis Cambridge University Press 1988 Vll

7 Preface 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 undergraduates. Th this end, this text addresses a number of issues that are rarely considered. For instance, when we apply the Laplace transform 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

8 X Preface Furthermore, it is sometimes desirable to take the Laplace transform 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. Another delicate problem in the literature has been the application of the Laplace transform to the so-called Dirac delta function. Except for texts on the theory of distributions, traditional treatments are usually heuristic in nature. In the present text we give a new and mathematically rigorous account of the Dirac delta function based upon the Riemann-Stieltjes integral. It is elementary in scope and entirely suited to this level of exposition. One of the highlights of the Laplace transform theory is the complex inversion formula, examined in Chapter 4. It is the most sophisticated tool in the Laplace transform arsenal. In order to facilitate understanding of the inversion formula and its many subsequent applications, a self-contained summary of the theory of complex variables is given in Chapter 3. On the whole, while setting out the theory as explicitly and carefully as possible, the wide range of practical applications for which the Laplace transform is so ideally suited also receive their due coverage. Thus I hope that the text will appeal to students of mathematics and engineering alike. Historical Summary. Integral transforms date back to the work of Leonard Euler (1763 and 1769), who considered them essentially in the form of the inverse Laplace transform in solving second-order, linear ordinary differential equations. Even Laplace, in his great work, Theorie analytique des probabilites (1812), credits Euler with introducing integral transforms. It is Spitzer (1878) who attached the name of Laplace to the expression employed by Euler. In this form it is substituted into the differential equation where y is the unknown function of the variable x. In the late 19th century, the Laplace transform was extended to its complex form by Poincare and Pincherle, rediscovered by Petzval,

9 Preface Xl and extended to two variables by Picard, with further investigations conducted by Abel and many others. The first application of the modern Laplace transform occurs in the work of Bateman (1910), who transforms equations arising from Rutherford's work on radioactive decay by setting p(x) = dp dt l I -=-AP 100 e-xtp(t)dt and obtaining the transformed equation. Bernstein (1920) used the expression f(s) = 100 e-su</j(u)du, calling it the Laplace transformation, in his work on theta functions. The modern approach was given particular impetus by Doetsch in the 1920s and 30s; he applied the Laplace transform to differential, integral, and integra-differential equations. This body of work culminated in his foundational1937 text, Theorie und Anwendungen der Laplace Transformation. No account of the Laplace transformation would be complete without mention of the work of Oliver Heaviside, who produced (mainly in the context of electrical engineering) a vast body of what is termed the "operational calculus!' This material is scattered throughout his three volumes, Electromagnetic Theory (1894, 1899, 1912), and bears many similarities to the Laplace transform method. Although Heaviside's calculus was not entirely rigorous, it did find favor with electrical engineers as a useful technique for solving their problems. Considerable research went into trying to make the Heaviside calculus rigorous and connecting it with the Laplace transform. One such effort was that of Bromwich, who, among others, discovered the inverse transform 1 1y+ioo X(t) = -. e 15 x(s)ds 2Jn y-ioo for y lying to the right of all the singularities of the function x.

10 Xll Preface Acknowledgments. Much of the Historical Summary has been taken from the many works of Michael Deakin of Monash University. I also wish to thank Alexander Krageloh for his careful reading of the manuscript and for his many helpful suggestions. I am also indebted to Aimo Hinkkanen, Sergei Federov, Wayne Walker, Nick Dudley Ward, and Allison Heard for their valuable input, to Lev Plimak for the diagrams, to Sione Ma'u for the answers to the exercises, and to Betty Fong for turning my scribbling into a text. Joel L. Schiff Auckland New Zealand

11 Contents Preface ix 35 1 Basic Principles The Laplace Thansform Convergence Continuity Requirements Exponential Order The Class,C Basic Properties of the Laplace Thansform Inverse of the Laplace Thansform Thanslation Theorems Differentiation and Integration of the Laplace Thansform Partial Fractions 0 2 Applications and Properties Gamma Function Periodic Functions o3 Derivatives Ordinary Differential Equations Dirac Operator Xlll

12 X1 V Contents 2.6 Asymptotic Values Convolution Steady-State Solutions 2.9 Difference Equations 3 Complex Variable Theory 3.1 Complex Numbers. 3.2 Functions Integration Power Series Integrals ofthe 'JYpe f~rx:j(x)dx. 4 Complex Inversion Formula 5 Partial Differential Equations Appendix References Th.bles Laplace Transform Operations Table of Laplace Transforms. Answers to Exercises Index