APPLIED MATHEMATICS APPLIED MATHEMATICS Gerald Dennis Mahan Pennsylvania State University University Park, Pennsylvania Springer Science+Business Media, LLC Library of Congress Cataloging-in-Publication Data Mahan, Gerald D. Applied mathematics!by Gerald D. Mahan. p. cm. Includes bibliographical references and index. ISBN 978-1-4613-5493-2 ISBN 978-1-4615-1315-5 (eBook) DOI 10.1007/978-1-4615-1315-5 1. Mathematics. 1. Title. QA37.3 .M342001 510-dc21 2001038769 ISBN 978-1-4613-5493-2 ©2002 Springer Science+Business Media New York Originally published by Kluwer Academic / Plenum Publishers, New York in 2002 Softcover reprint ofthe hardcover lst edition 2002 http://www.wkap.nl ill 9 8 7 6 5 4 3 2 1 A C.I.P. record for this book is available from the Library of Congress AII rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permis sion from the Publisher Preface This volume is a textbook for a year-long graduate level course in applied mathematics for scientists and engineers. All research universities have such a course, which could be taught in different departments, such as mathematics, physics, or engineering. I volunteered to teach this course when I realized that my own research students did not learn much in this course at my university. Then I learned that the available textbooks were too introduc tory. While teaching this course without an assigned text, I wrote up my lecture notes and gave them to the students. This textbook is a result of that endeavor. When I took this course many, many, years ago, the primary references were the two volumes of P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953). The present text returns the contents to a similar level, although the syllabus is quite different than given in this venerable pair of books. My own research field is theoretical condensed matter physics. My first professional job was at the Research and Development Center of the General Electric Company. When I left there to become a college professor, they hired me back as a consultant. For many years I wrote analytic and computer models of all possible semiconductor devices for GE-CRD. In this latter activity I really learned applied mathematics. Most often I was solving the diffusion equation. Heat diffuses, as do minority carriers in devices. I once described consulting as "solving the diffusion equations in all possible dimen sions with all possible initial and boundary conditions." Solving the transient diffusion equation is heavily emphasised in the chapters on partial differential equations. Similarly, the Laplace transform is most useful when solving transient problems, and is readily applied to diffusion. Conformal mapping is extraordinarily useful for solving problems in two dimensions, so this topic also gets emphasis. Most of the syllabus is traditional. There are chapters on matrices, group theory, special functions, complex variables, and linear and nonlinear differential equations. Problems, which can be assigned as homework, can be found at the end of each chapter. There are several sections not found in most textbooks. Wavelets are included in the chapter on transforms, along with those of Fourier and Laplace. Wavelets are a new topic that have emerged in the past twenty years. It is applied to data streams that never end in time. v vi Another unusual topic is the method of Markov averaging of random systems. The solution proposed by S. Chandrasekhar is useful for many different types of problems. PREFACE I wish to thank my students in this course, who used the notes which became the preliminary draft. They let me know quickly if the first version was unclear. Also, I thank my editor at Kluwer-Academic-Plenum, Amelia McNamara, who agreed to publish another of my manuscripts. Her staff are always pleasant and efficient. Contents 1. Determinants .......................................................... . 1 1.1. Cramer's Rule ..................................................... . 3 1.2. Gaussian Elimination ............................................. . 4 1.3. Special Determinants ............................................. . 7 2. Matrices ............................................................... . 13 2.1. Several Theorems ................................................. . 14 2.2. Linear Equations .................................................. . 15 2.3. Inverse of a Matrix ................................................ . 17 2.4. Eigenvalues and Eigenvectors .................................... . 18 2.5. Unitary Transformations ......................................... . 23 2.6. Hon-Hermitian Matrices ......................................... . 25 2.7. A Special Matrix .................................................. . 32 2.8. Gram-Schmidt .................................................... . 34 2.9. Chains ............................................................. . 38 3. Group Theory ......................................................... . 47 3.1. Basic Properties of Groups ....................................... . 47 3.2. Group Representations ........................................... . 51 3.3. Characters ......................................................... . 54 3.4. Direct Product Groups ........................................... . 58 3.4.1. Representations ............................................ . 58 3.5. Basis Functions ................................................... . 60 3.6. Angular Momentum .............................................. . 61 3.7. Products of Representations ...................................... . 64 3.8. Quantum Mechanics .............................................. . 64 3.8.1. Eigenvalues ................................................. . 64 3.8.2. Representations ............................................ . 65 3.8.3. Matrix Elements ........................................... . 67 3.9. Double Groups .................................................... . 68 4. Complex Variables ................................................... . 73 4.1. Introduction ....................................................... . 73 vii viii 4.2. Analytic Functions................................................. 75 4.3. Multivalued Functions. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4. Contour Integrals .................................................. 84 CONTENTS 4.5. Meromorphic Functions ........................................... 99 4.6. Higher Poles.... .................................................... 101 4.7. Integrals Involving Branch Cuts .................................. 102 4.8. Approximate Evaluation of Integrals ............................. 109 4.8.1. Steepest Descent ............................................ 109 4.8.2. Saddle Point Integrals ...................................... 110 5. Series ................................................................... 119 5.1. Taylor Series ....................................................... 119 5.2. Convergence ........................................................ 121 5.3. Laurent Series ...................................................... 126 5.4. Meromorphic Functions ........................................... 128 5.5. Asymptotic Series .................................................. 129 5.6. Summing Series .................................................... 132 5.7. Pade Approximants ................................................ 135 6. Conformal Mapping................................... ................ 141 6.1. Laplace's Equation................................................. 141 6.2. Mapping................... .... ...................... ..... ...... .... 144 6.3. Examples ........................................................... 149 6.4. Schwartz-Christoffel Transformations............................ 157 6.5. van der Pauw ....................................................... 167 6.5.1. Currents ..................................................... 168 6.5.2. Resistance ................................................... 170 6.5.3. Mapping ..................................................... 172 7. Markov Averaging..................................................... 177 7.1. Random Walk.. .................................................... 177 7.2. Speckle.............................................................. 183 7.3. Inhomogeneous Broadening....................................... 188 8. Fourier Transforms ................................................... 195 8.1. Fourier Transforms ................................................ 195 8.1.1. Unbounded Space........................................... 195 8.1.2. Half-Space................................................... 197 8.1.3. Finite Systems in 1D ........................................ 199 8.2. Laplace Transforms ................................................ 202 8.3. Wavelets............................................................ 205 8.3.1. Continous Wavelet Transform ............................. 207 8.3.2. Discrete Transforms ........................................ 210 9. Equations of Physics ................................................. 217 9.1. Boundary and Initial Conditions.................................. 218 9.2. Boltzmann Equation............................................... 218 9.2.1. Moment Equations ......................................... 219 9.2.2. Diffusion Equations ......................................... 223 9.2.3. Fluid Equations ............................................ . 223 ix 9.3. Solving Differential Equations .................................... . 225 9.3.1. Homogeneous Linear Equations .......................... . 225 CONTENTS 9.3.2. Inhomogeneous Linear Equations ........................ . 227 9.3.3. Nonlinear Equations ....................................... . 229 9.4. Elliptic Integrals ................................................... . 235 10. One Dimension ....................................................... . 237 10.1. Introduction ...................................................... . 237 10.2. Diffusion Equation ............ , .................................. . 240 10.3. Wave Equation ............... , .................................. . 252 11. Two Dimensions ..................................................... . 265 11.1. Rectangular Coordinates ........................................ . 265 11.1.1. Laplace's Equation ...................................... . 265 11.1.2. Diffusion Equation ...................................... . 268 11.1.3. Wave Equation .......................................... . 272 11.2. Polar Coordinates ............................................... . 273 11.2.1. Laplace's Equation ...................................... . 274 11.2.2. Helmholtz Equation ..................................... . 279 11.2.3. Hankel Transforms ...................................... . 286 12. Three Dimensions .................................................... . 297 12.1. Cartesian Coordinates ........................................... . 297 12.2. Cylindrical Coordinates ......................................... . 299 12.3. Spherical Coordinates ........................................... . 305 12.3.1. Laplace's Equation ...................................... . 306 12.3.2. Diffusion and Wave Equations ......................... . 309 12.4. Problems Inside a Sphere ........................................ . 315 12.5. Vector Wave Equation .......................................... . 319 12.5.1. Bulk Waves .............................................. . 319 12.5.2. Boundary Conditions ................................... . 320 13. Odds and Ends ....................................................... . 333 13.1. Hypergeometric Functions ...................................... . 333 13.1.1. Continued Fractions .................................... . 337 13.1.2. Solving Equations With Series .......................... . 339 13.2. Orthogonal Polynomials ........................................ . 343 13.2.1. Parabolic Cylinder Functions ........................... . 344 13.2.2. Hermite Polynomials .................................... . 344 13.2.3. Laguerre Polynomials ................................... . 346 13.3. Sturm-Liouville ................................................. . 347 13.4. Green's Functions ............................................... . 352 13.5. Singular Integral Equations ..................................... . 357 Index ........................................................................ . 365 Applied Mathematics 1 Determ in ants A determinant is a single number obtained by a particular evaluation of an n x n matrix of values. The number is real if the elements of the matrix are real, but could be complex if the elements are complex. Below are given some techniques for evaluating the determinant of a n x n matrix. However, it is best to give some simple examples of matrix of small dimension: 1. n = 1: Here the matrix has a single element, and the determinant is the value of that number. 2. n = 2: Here the matrix has four elements, and the determinant is: (1.1) The two off-diagonal elements are multiplied together, and subtracted from the product of the diagonal elements. 3. n = 3: Here there are nine elements in the matrix. The determinant is all a12 a13 det lal = det a21 a22 a23 a31 a32 a33 = + + alla22a33 a12a23a31 a13a32a21 (1.2) There are six terms: three have plus signs and three have minus signs. These three cases are the only ones that can be found in a simple fashion. For larger values of n it is necessary to employ minors. Minors The minor Mij of an element aij of a determinant is the determinant 1 one gets by crossing out row i and columnj. The minor has dimension (n - 1) G. D. Mahan, Applied Mathematics © Kluwer Academic / Plenum Publishers, New York 2002