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MODERN MATHEMATICAL METHODS FOR
PHYSICISTS AND ENGINEERS
The advent of powerful desktop computers has revolutionized scientific analysis and
engineering design in fields as disparate as particle physics and telecommunications.
Modern Mathematical Methods for Physicists and Engineers provides an up-to-date
mathematical and computational education for students, researchers, and practicing
engineers.
The author begins with a review of computation and then deals with a range of
key concepts including sets, fields, matrix theory, and vector spaces. He then goes on
to cover more advanced subjects such as linear mappings, group theory, and special
functions. Throughout, he concentrates exclusively on the most important topics for the
working physical scientist or engineer, with the aim of helping them to make intelligent
use of the latest computational and analytical methods.
The book contains well over 400 homework problems and covers many topics not
dealt with in other textbooks. It will be an ideal textbook for senior undergraduate
and graduate students in the physical sciences and engineering, as well as a valuable
reference for working engineers.
C. D. Cantrell received his Ph.D. from Princeton University in 1968. He taught at
Swarthmore College from 1967 until 1973 and was a staff member at the Los Alamos
National Laboratory from 1973 until 1979. Since then he has been at the University
of Texas at Dallas, where he is Professor of Physics and Electrical Engineering, and
Director of the Photonic Technology and Engineering Center. Professor Cantrell is a
consultant for Alcatel USA and Ericsson and is a Fellow of the American Physical
Society, the Optical Society of America, and the IEEE.
MODERN
MATHEMATICAL
METHODS FOR
PHYSICISTS AND
ENGINEERS
C. D. CANTRELL
CAMBRIDGE
UNIVERSITY PRESS
; '> J. 4ara Library
JIlKt UNIVERSITY
^rtlif^OOOK, ONTARfO
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk
40 West 20th Street, New York, NY 10011-4211, USA www.cup.org
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
Ruiz de Alarcon 13, 28014 Madrid, Spain
© C. D. Cantrell 2000
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2000
Printed in the United States of America
Typeface Times Roman 10.5/13 pt. System UTgX2£ [tb]
A catalog record for this book is available from
the British Library.
Library of Congress Cataloging in Publication Data
Cantrell, C. D. (Cyrus D.), 1940-
Modern mathematical methods for physicists and engineers / C. D.
Cantrell.
p. cm.
Includes bibliographical references (p. - ).
ISBN 0-521 -59180-5 (hb). - ISBN 0-521 -59827-3 (pbk.)
1. Mathematics. I. Title.
QA37.2.C24 1999
510 - dc21 98-24761
CIP
ISBN 0 521 59180 5 hardback
ISBN 0 521 59827 3 paperback
iVlUU
* I
To Lynn, Kate, and Sarah
Digitized by the Internet Archive
in 2019 with funding from
Kahle/Austin Foundation
https://archive.org/details/modernmathematicOOOOcant
CONTENTS
Preface page xix
1 FOUNDATIONS OF COMPUTATION 1
1.1 Introduction 1
1.2 Representations of Numbers 2
1.2.1 Integers 3
1.2.2 Rational Numbers and Real Numbers 14
1.2.3 Representations of Numbers as Text 17
1.2.4 Exercises for Section 1.2 20
1.3 Finite Floating-point Representations 21
1.3.1 Simple Cases 21
1.3.2 Practical Floating-point Representations 25
1.3.3 Approaching Zero or Infinity Gracefully 28
1.3.4 Exercises for Section 1.3 30
1.4 Floating-point Computation 31
1.4.1 Relative Error; Machine Epsilon 31
1.4.2 Rounding 32
1.4.3 Floating-point Addition and Subtraction 35
1.4.4 Exercises for Section 1.4 36
1.5 Propagation of Errors 37
1.5.1 General Formulas 37
1.5.2 Examples of Error Propagation 39
1.5.3 Estimates of the Mean and Variance 41
1.5.4 Exercises for Section 1.5 43
1.6 Bibliography and Endnotes 45
1.6.1 Bibliography 45
1.6.2 Endnotes 46
2 SETS AND MAPPINGS 47
2.1 Introduction 47
2.2 Basic Definitions 49
2.2.1 Sets 49
2.2.2 Mappings 53
2.2.3 Axiom of Choice 62
2.2.4 Cartesian Products 62
vii
viii CONTENTS
2.2.5 Equivalence and Equivalence Classes 65
2.2.6 Exercises for Section 2.2 67
2.3 Union, Intersection, and Complement 68
2.3.1 Unions of Sets 68
2.3.2 Intersections of Sets 69
2.3.3 Relative Complement 70
2.3.4 De Morgan’s Laws 71
2.3.5 Exercises for Section 2.3 71
2.4 Infinite Sets 72
2.4.1 Basic Properties of Infinite Sets 72
2.4.2 Induction and Recursion 73
2.4.3 Countable Sets 76
2.4.4 Countable Unions and Intersections 77
2.4.5 Uncountable Sets 78
2.4.6 Exercises for Section 2.4 80
2.5 Ordered and Partially Ordered Sets 82
2.5.1 Partial Orderings 82
2.5.2 Orderings; Upper and Lower Bounds 83
2.5.3 Maximal Chains 84
2.5.4 Exercises for Section 2.5 84
2.6 Bibliography 85
EVALUATION OF FUNCTIONS 86
3.1 Introduction 86
3.2 Sensitivity and Condition Number 86
3.2.1 Definitions 86
3.2.2 Evaluation of Polynomials 87
3.2.3 Multiple Roots of Polynomials 89
3.2.4 Exercises for Section 3.2 91
3.3 Recursion and Iteration 92
3.3.1 Finding Roots by Bisection 92
3.3.2 Newton-Raphson Method 92
3.3.3 Evaluation of Series 95
3.3.4 Exercises for Section 3.3 97
3.4 Introduction to Numerical Integration 99
3.4.1 Rectangle Rules 101
3.4.2 Trapezoidal Rule 102
3.4.3 Local and Global Errors 102
3.4.4 Exercises for Section 3.4 105
3.5 Solution of Differential Equations 106
3.5.1 Euler’s Method 107
3.5.2 Truncation Error of Euler’s Method 109
3.5.3 Stability Analysis of Euler’s Method 111
