Fourier Transforms Laplace Transforms * γ = Euler’s constant 0.577215665. FOUNDATIONS OF APPLIED MATHEMATICS Michael D. Greenberg Professor Emeritus Department of Mechanical and Aerospace Engineering University of Delaware Dover Publications, Inc. Mineola, New York Copyright Copyright © 1978 by Michael D. Greenberg All rights reserved. Bibliographical Note This Dover edition, first published in 2013, is an unabridged republication of the work originally published by Prentice-Hall, Inc., Englewood Cliffs, N.J., in 1978. This book was previously published by Pearson Education, Inc. Library of Congress Cataloging-in-Publication Data Greenberg, Michael D., 1935–author. Foundations of applied mathematics / Michael D. Greenberg, professor emeritus, Department of Mechanical and Aerospace Engineering, University of Delaware. – Dover edition. p. cm. Summary: “A longtime classic text in applied mathematics, this volume also serves as a reference for undergraduate and graduate students of engineering. Topics include real variable theory, complex variables, linear analysis, partial and ordinary differential equations, and other subjects Answers to selected exercises are provided, along with Fourier and Laplace transformation tables and useful formulas. 1978 edition”—Provided by publisher. Includes bibliographical references and index. eISBN-13: 978-0-486-78218-8 1. Engineering mathematics. I. Title. TA330.G73 2013 620.001'51—dc23 2013015816 Manufactured in the United States by Courier Corporation 49279601 2013 www.doverpublications.com This book is dedicated, with love, to Mim. Contents by Part Part I Real Variable Theory II Complex Variables III Linear Analysis IV Ordinary Differential Equations V Partial Differential Equations Contents Preface PART I REAL VARIABLE THEORY Chapter 1 The Important Limit Processes 1.1. Functions and Functionals 1.2. Limits, Continuity, and Uniform Continuity 1.3. Differentiation 1.4. Integration 1.5. Asymptotic Notation and the “Big Oh” 1.6. Numerical Integration 1.7. Differentiation of Integrals Containing a Parameter; Leibnitz’s Rule Exercises Chapter 2 Infinite Series 2.1. Sequences and Series; Fundamentals and Tests for Convergence 2.2. Series with Terms of Mixed Sign 2.3. Series with Terms That Are Functions; Power Series 2.4. Taylor Series 2.5. That’s All Very Nice, But How Do We Sum the Series? Acceleration Techniques *2.6. Asymptotic Expansions Exercises Chapter 3 Singular Integrals 3.1. Choice of Summability Criteria; Convergence and Cauchy Principal Value 3.2. Tests for Convergence 3.3. The Gamma Function 3.4. Evaluation of Singular Integrals Exercises Chapter 4 Interchange of Limit Processes and the Delta Function 4.1. Theorems on Limit Interchange 4.2. The Delta Function and Generalized Functions Exercises Chapter 5 Fourier Series and the Fourier Integral 5.1. The Fourier Series of f(x) 5.2. Pointwise Convergence of the Series 5.3. Termwise Integration and Differentiation of Fourier Series 5.4. Variations : Periods Other than 2π, and Finite Interval 5.5. Infinite Period ; the Fourier Integral Exercises Chapter 6 Fourier and Laplace Transforms 6.1. The Fourier Transform 6.2. The Laplace Transform 6.3. Other Transforms Exercises