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Mathematical Methods for Engineers and Scientists 1 K.T. Tang Mathematical Methods for Engineers and Scientists 1 Complex Analysis, Determinants and Matrices With 49 Figures and 2 Tables 123 Professor Dr. Kwong-Tin Tang Paciﬁc Lutheran University Department of Physics Tacoma, WA 98447, USA E-mail: tangka@plu.edu Library of Congress Control Number: 2006932619 ISBN-10 3-540-30273-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30273-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A X macro package Typesetting by the author and SPi using a Springer LT E Cover design: eStudio Calamar Steinen Printed on acid-free paper SPIN: 11576396 57/3100/SPi 543210 Preface For some 30 years, I have taught two "Mathematical Physics" courses. One of them was previously named "Engineering Analysis." There are several textbooks of unquestionable merit for such courses, but I could not ﬁnd one that ﬁtted our needs. It seemed to me that students might have an easier time if some changes were made in these books. I ended up using class notes. Actually, I felt the same about my own notes, so they got changed again and again. Throughout the years, many students and colleagues have urged me to publish them. I resisted until now, because the topics were not new and I was not sure that my way of presenting them was really much better than others. In recent years, some former students came back to tell me that they still found my notes useful and looked at them from time to time. The fact that they always singled out these courses, among many others I have taught, made me think that besides being kind, they might even mean it. Perhaps it is worthwhile to share these notes with a wider audience. It took far more work than expected to transcribe the lecture notes into printed pages. The notes were written in an abbreviated way without much explanation between any two equations, because I was supposed to supply the missing links in person. How much detail I would go into depended on the reaction of the students. Now without them in front of me, I had to decide the appropriate amount of derivation to be included. I chose to err on the side of too much detail rather than too little. As a result, the derivation does not look very elegant, but I also hope it does not leave any gap in students' comprehension. Precisely stated and elegantly proved theorems looked great to me when I was a young faculty member. But in later years, I found that elegance in the eyes of the teacher might be stumbling blocks for students. Now I am convinced that before the student can use a mathematical theorem with conﬁdence, he or she must ﬁrst develop an intuitive feeling. The most eﬀective way to do that is to follow a suﬃcient number of examples. This book is written for students who want to learn but need a ﬁrm handholding. I hope they will ﬁnd the book readable and easy to learn from. VI Preface Learning, as always, has to be done by the student herself or himself. No one can acquire mathematical skill without doing problems, the more the better. However, realistically students have a ﬁnite amount of time. They will be overwhelmed if problems are too numerous, and frustrated if problems are too diﬃcult. A common practice in textbooks is to list a large number of problems and let the instructor to choose a few for assignments. It seems to me that is not a conﬁdence building strategy. A self-learning person would not know what to choose. Therefore a moderate number of not overly diﬃcult problems, with answers, are selected at the end of each chapter. Hopefully after the student has successfully solved all of them, he or she will be encouraged to seek more challenging ones. There are plenty of problems in other books. Of course, an instructor can always assign more problems at levels suitable to the class. On certain topics, I went farther than most other similar books, not in the sense of esoteric sophistication, but in making sure that the student can carry out the actual calculation. For example, the diagonalization of a degenerate hermitian matrix is of considerable importance in many ﬁelds. Yet to make it clear in a succinct way is not easy. I used several pages to give a detailed explanation of a speciﬁc example. Professor I.I. Rabi used to say "All textbooks are written with the principle of least astonishment." Well, there is a good reason for that. After all, textbooks are supposed to explain away the mysteries and make the profound obvious. This book is no exception. Nevertheless, I still hope the reader will ﬁnd something in this book exciting. This volume consists of three chapters on complex analysis and three chapters on theory of matrices. In subsequent volumes, we will discuss vector and tensor analysis, ordinary diﬀerential equations and Laplace transforms, Fourier analysis and partial diﬀerential equations. Students are supposed to have already completed two or three semesters of calculus and a year of college physics. This book is dedicated to my students. I want to thank my A and B students, their diligence and enthusiasm have made teaching enjoyable and worthwhile. I want to thank my C and D students, their diﬃculties and mistakes made me search for better explanations. I want to thank Brad Oraw for drawing many ﬁgures in this book, and Mathew Hacker for helping me to typeset the manuscript. I want to express my deepest gratitude to Professor S.H. Patil, Indian Institute of Technology, Bombay. He has read the entire manuscript and provided many excellent suggestions. He has also checked the equations and the problems and corrected numerous errors. Without his help and encouragement, I doubt this book would have been. The responsibility for remaining errors is, of course, entirely mine. I will greatly appreciate if they are brought to my attention. Tacoma, Washington October 2005 K.T. Tang Contents Part I Complex Analysis 1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Our Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Addition and Multiplication of Integers . . . . . . . . . . . . . . 1.1.2 Inverse Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Fractional Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Napier's Idea of Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Briggs' Common Logarithm . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Peculiar Number Called e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Unique Property of e . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Natural Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Approximate Value of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Exponential Function as an Inﬁnite Series . . . . . . . . . . . . . . 1.4.1 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 The Limiting Process Representing e . . . . . . . . . . . . . . . . . 1.4.3 The Exponential Function ex . . . . . . . . . . . . . . . . . . . . . . . 1.5 Uniﬁcation of Algebra and Geometry . . . . . . . . . . . . . . . . . . . . . . 1.5.1 The Remarkable Euler Formula . . . . . . . . . . . . . . . . . . . . . 1.5.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Polar Form of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Powers and Roots of Complex Numbers . . . . . . . . . . . . . . 1.6.2 Trigonometry and Complex Numbers . . . . . . . . . . . . . . . . 1.6.3 Geometry and Complex Numbers . . . . . . . . . . . . . . . . . . . 1.7 Elementary Functions of Complex Variable . . . . . . . . . . . . . . . . . 1.7.1 Exponential and Trigonometric Functions of z . . . . . . . . 3 3 4 5 6 7 8 9 13 13 15 18 18 19 21 21 21 23 24 24 24 25 28 30 33 40 46 46 VIII Contents 1.7.2 1.7.3 1.7.4 Exercises Hyperbolic Functions of z . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithm and General Power of z . . . . . . . . . . . . . . . . . . Inverse Trigonometric and Hyperbolic Functions . . . . . . . ................................................... 48 50 55 58 2 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1.1 Complex Function as Mapping Operation . . . . . . . . . . . . 62 2.1.2 Diﬀerentiation of a Complex Function . . . . . . . . . . . . . . . . 62 2.1.3 Cauchy–Riemann Conditions . . . . . . . . . . . . . . . . . . . . . . . 65 2.1.4 Cauchy–Riemann Equations in Polar Coordinates . . . . . 67 2.1.5 Analytic Function as a Function of z Alone . . . . . . . . . . . 69 2.1.6 Analytic Function and Laplace's Equation . . . . . . . . . . . . 74 2.2 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2.1 Line Integral of a Complex Function . . . . . . . . . . . . . . . . . 81 2.2.2 Parametric Form of Complex Line Integral . . . . . . . . . . . 84 2.3 Cauchy's Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.3.1 Green's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.3.2 Cauchy–Goursat Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.3.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . 90 2.4 Consequences of Cauchy's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 93 2.4.1 Principle of Deformation of Contours . . . . . . . . . . . . . . . . 93 2.4.2 The Cauchy Integral Formula . . . . . . . . . . . . . . . . . . . . . . . 94 2.4.3 Derivatives of Analytic Function . . . . . . . . . . . . . . . . . . . . 96 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3 Complex Series and Theory of Residues . . . . . . . . . . . . . . . . . . 107 3.1 A Basic Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1 The Complex Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.2 Convergence of Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2.4 Uniqueness of Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.3 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.3.1 Uniqueness of Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . 120 3.4 Theory of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.1 Zeros and Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.2 Deﬁnition of the Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.4.3 Methods of Finding Residues . . . . . . . . . . . . . . . . . . . . . . . 129 3.4.4 Cauchy's Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.4.5 Second Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.5 Evaluation of Real Integrals with Residues . . . . . . . . . . . . . . . . . . 141 3.5.1 Integrals of Trigonometric Functions . . . . . . . . . . . . . . . . . 141 3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . 144 Contents IX 3.5.3 Fourier Integral and Jordan's Lemma . . . . . . . . . . . . . . . . 147 3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-shaped Contour . . . . . . . . . . . . 153 3.5.5 Integration Along a Branch Cut . . . . . . . . . . . . . . . . . . . . . 158 3.5.6 Principal Value and Indented Path Integrals . . . . . . . . . . 160 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Part II Determinants and Matrices 4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.1.1 Solution of Two Linear Equations . . . . . . . . . . . . . . . . . . . 173 4.1.2 Properties of Second-Order Determinants . . . . . . . . . . . . . 175 4.1.3 Solution of Three Linear Equations . . . . . . . . . . . . . . . . . . 175 4.2 General Deﬁnition of Determinants . . . . . . . . . . . . . . . . . . . . . . . . 179 4.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.2.2 Deﬁnition of a nth Order Determinant . . . . . . . . . . . . . . . 181 4.2.3 Minors, Cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.2.4 Laplacian Development of Determinants by a Row (or a Column) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.4 Cramer's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.4.1 Nonhomogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.4.2 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 4.5 Block Diagonal Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.6 Laplacian Developments by Complementary Minors . . . . . . . . . . 198 4.7 Multiplication of Determinants of the Same Order . . . . . . . . . . . 202 4.8 Diﬀerentiation of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.9 Determinants in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.1 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.1.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.1.2 Some Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.1.3 Matrix Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 5.1.4 Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.2.1 Product of Two Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.2.2 Motivation of Matrix Multiplication . . . . . . . . . . . . . . . . . 223 5.2.3 Properties of Product Matrices . . . . . . . . . . . . . . . . . . . . . . 225 5.2.4 Determinant of Matrix Product . . . . . . . . . . . . . . . . . . . . . 230 5.2.5 The Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 X Contents 5.3 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.3.1 Gauss Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.3.2 Existence and Uniqueness of Solutions of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.4 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.4.1 Nonsingular Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.4.2 Inverse Matrix by Cramer's Rule . . . . . . . . . . . . . . . . . . . . 243 5.4.3 Inverse of Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . 246 5.4.4 Inverse Matrix by Gauss–Jordan Elimination . . . . . . . . . 248 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6 Eigenvalue Problems of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1.1 Secular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1.2 Properties of Characteristic Polynomial . . . . . . . . . . . . . . 262 6.1.3 Properties of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.2 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 6.2.1 Hermitian Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 6.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.2.3 Gram–Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.3 Unitary Matrix and Orthogonal Matrix . . . . . . . . . . . . . . . . . . . . 271 6.3.1 Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 6.3.2 Properties of Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . 272 6.3.3 Orthogonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 6.3.4 Independent Elements of an Orthogonal Matrix . . . . . . . 274 6.3.5 Orthogonal Transformation and Rotation Matrix . . . . . . 275 6.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 6.4.1 Similarity Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 278 6.4.2 Diagonalizing a Square Matrix . . . . . . . . . . . . . . . . . . . . . . 281 6.4.3 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 6.5 Hermitian Matrix and Symmetric Matrix . . . . . . . . . . . . . . . . . . . 286 6.5.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.5.2 Eigenvalues of Hermitian Matrix . . . . . . . . . . . . . . . . . . . . 287 6.5.3 Diagonalizing a Hermitian Matrix . . . . . . . . . . . . . . . . . . . 288 6.5.4 Simultaneous Diagonalization . . . . . . . . . . . . . . . . . . . . . . . 296 6.6 Normal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.7 Functions of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 6.7.1 Polynomial Functions of a Matrix . . . . . . . . . . . . . . . . . . . 300 6.7.2 Evaluating Matrix Functions by Diagonalization . . . . . . . 301 6.7.3 The Cayley–Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . 305 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Part I Complex Analysis

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