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LAPLACE TRANSFORMS

MURRAY R. SPIEGEL, Ph. D.

Laplace transforms applications completely explainedv Works with all major texts

450 fully solved problems

Perfect for brushup or exam prep

Use with these courses: Operational Calculus 9 Electrical Engineering

9 Mechanics RT College Ikthematics

SCHAUM'S OUTLINE OF

THEORY AND PROBLEMS

OF

LAPLACE TRANSFORMS

.

MURRAY R. SPIEGEL, Ph.D. Former Professor and Chairman.

Mathematics Department

Rensselaer Polytechnic Institute

Hartford Graduate Center

.

SCHAUM'S OUTLINE SERIES McGRAW-HILL

New York San Francisco Washingtun. D.C. Auckland Rogoid Caracas Lisbon London Madrid Alexien City ,Milan Aluntrcal New Delhi

San Juan Singapore Sydney lbkva "lomnro

Copyright © 1965 by McGraw-Hill. Inc. All Rights Reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval -system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

60231

12131415S11SH754321069

Preface

The theory of Laplace transforms or Laplace transformation, also referred to as operational calculus, has in recent years become an essential part of the mathematical background required of engineers, physicists, mathematicians and other scientists. This is because, in addition to being of great theoretical interest in itself, Laplace transform methods provide easy and effective means for the solution of many problems arising in various fields of science and engineering.

The subject originated in attempts to justify rigorously certain "operational rules" used by Heaviside in the latter part of the 19th century for solving equations in electro- magnetic theory. These attempts finally proved successful in the early part of the 20th century through the efforts of Bromwich, Carson, van der Pol and other mathematicians who employed complex variable theory.

This book is designed for use as a supplement to all current standard texts or as a textbook for a formal course in Laplace transform theory and applications. It should also be of considerable value to those taking courses in mathematics, physics, electrical engi- neering, mechanics, heat flow or any of the numerous other fields in which Laplace transform methods are employed.

Each chapter begins with a clear statement of pertinent definitions, principles and theorems together with illustrative and other descriptive material. This is followed by graded sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory, bring into sharp focus those fine points without which the student continually feels himself on unsafe ground, and provide the repetition of basic principles so vital to effective learning. Numerous proofs of theorems and derivations of formulas are included among the solved problems. The large number of supplementary problems with answers serve as a complete review of the material in each chapter.

Topics covered include the properties of Laplace transforms and inverse Laplace transforms together with applications to ordinary and partial differential equations, integral equations, difference equations and boundary-value problems. The theory using complex variables is not treated until the last half of the book. This is done, first, so that the student may comprehend and appreciate more fully the theory, and the power, of the complex inversion formula and, second, to meet the needs of those who wish only an introduction to the subject. Chapters on complex variable theory and Fourier series and integrals, which are important in a discussion of the complex inversion formula, have been included for the benefit of those unfamiliar with these topics.

Considerably more material has been included here than can be covered in most first courses. This has been done to make the book more flexible, to provide a more useful book of reference and to stimulate further interest in the topics.

I wish to take this opportunity to thank the staff of the Schaum Publishing Company for their splendid cooperation.

M. R. SPIEGEL Rensselaer Polytechnic Institute January, 1965

CONTENTS

Page

Chapter THE LAPLACE TRANSFORM ................................ 1 Definition of the Laplace transform. Notation. Laplace transforms of some elementary functions. Sectional or piecewise continuity. Functions of ex- ponential order. Sufficient conditions for existence of Laplace transforms. Some important properties of Laplace transforms. Linearity property. First translation or shifting property. Second translation or shifting property. Change of scale property. Laplace transform of derivatives. Laplace trans- form of integrals. Multiplication by tn. Division by t. Periodic functions. Behavior of f (s) as s--. Initial-value theorem. Final-value theorem. Generalization of initial-value theorem. Generalization of final-value theorem. Methods of finding Laplace transforms. Direct method. Series method. Method of differential equations. Differentiation with respect to a parameter. Miscellaneous methods. Use of Tables. Evaluation of integrals. Some special functions. The gamma function. Bessel functions. The error function. The complementary error function. Sine and cosine integrals. Exponential integral. Unit step function. Unit impulse or Dirac delta function. Null functions. Laplace transforms of special functions.

1

Chapter 2

Chapter 3

THE INVERSE LAPLACE TRANSFORM .................... 42 Definition of inverse Laplace transform. Uniqueness of inverse Laplace trans- forms. Lerch's theorem. Some inverse Laplace transforms. Some important properties of inverse Laplace transforms. Linearity property. First transla- tion or shifting property. Second translation or shifting property. Change of scale property. Inverse Laplace transform of derivatives. Inverse Laplace transform of integrals. Multiplication by sn. Division by s. The convolution property. Methods of finding inverse Laplace transforms. Partial fractions method. Series methods. Method of differential equations. Differentiation with respect to a parameter. Miscellaneous methods. Use of Tables. The complex inversion formula. The Heaviside expansion formula. The beta function. Evaluation of integrals.

APPLICATIONS TO DIFFERENTIAL EQUATIONS.......... 78 Ordinary differential equations with constant coefficients. Ordinary differen- tial equations with variable coefficients. Simultaneous ordinary differential equations. Applications to mechanics. Applications to electrical circuits. Applications to beams. Partial differential equations.

Chapter 4 APPLICATIONS TO INTEGRAL AND DIFFERENCE EQUATIONS .................................. Integral equations. Integral equations of convolution type. Abel's integral equation. The tautochrone problem. Integro-differential equations. Difference equations. Differential-difference equations.

Chapter 5 COMPLEX VARIABLE THEORY ............................. The complex number system. Polar form of complex numbers. Operations in polar form. De Moivre's theorem. Roots of complex numbers. Functions. Limits and continuity. Derivatives. Cauchy-Riemann equations. Line in- tegrals. Green's theorem in the plane. Integrals. Cauchy's theorem. Cauchy's integral formulas. Taylor's series. Singular points. Poles. Laurent's series. Residues. Residue theorem. Evaluation of definite integrals.

112

136

CONTENTS

Chapter 6 FOURIER SERIES AND INTEGRALS ........................ Fourier series. Odd and even functions. Half range Fourier sine and cosine series. Complex form of Fourier series. Parseval's identity for Fourier series. Finite Fourier transforms. The Fourier integral. Complex form of Fourier integrals. Fourier transforms. Fourier sine and cosine transforms. The convolution theorem. Parseval's identity for Fourier integrals. Relationship of Fourier and Laplace transforms.

Page

173

Chapter 7 THE COMPLEX INVERSION FORMULA ................... 201 The complex inversion formula. The Bromwich contour. Use of residue theorem in finding inverse Laplace transforms. A sufficient condition for the integral around r to approach zero. Modification of Bromwich contour in case of branch points. Case of infinitely many singularities.

Chapter 8 APPLICATIONS TO BOUNDARY-VALUE PROBLEMS....... 219 Boundary-value problems involving partial differential equations. Some im- portant partial differential equations. One dimensional heat conduction equa- tion. One dimensional wave equation. Longitudinal vibrations of a beam. Transverse vibrations of a beam. Heat conduction in a cylinder. Transmission lines. Two and three dimensional problems. Solution of boundary-value problems by Laplace transforms.

APPENDIX A. TABLE OF GENERAL PROPERTIES OF LAPLACE TRANSFORMS ................................. 243

APPENDIX B. TABLE OF SPECIAL LAPLACE TRANSFORMS.......... 245

APPENDIX C. TABLE OF SPECIAL FUNCTIONS ....................... 255

INDEX ...................................................................... 257

Chapter 1

The Laplace Transform

DEFINITION OF THE LAPLACE TRANSFORM Let F(t) be a function of t specified for t> 0. Then the Laplace transform of F(t),

denoted by 4 (F(t)), is defined by

{F(t)) = f(s) = f e-St F(t) dt (1) 0

where we assume at present that'the parameter s is real. Later it will be found useful to consider s complex.

The Laplace transform of F(t) is sai