Arfken Mathematical Methods For Physicists 6ed Pdf

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1. MATHEMATICAL METHODS FOR PHYSICISTS SIXTH EDITION George B. Arfken Miami University Oxford, OH Hans J. Weber University of Virginia Charlottesville, VA Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo 2. This page intentionally left blank 3. MATHEMATICAL METHODS FOR PHYSICISTS SIXTH EDITION 4. This page intentionally left blank 5. Acquisitions Editor Tom Singer Project Manager Simon Crump Marketing Manager Linda Beattie Cover Design Eric DeCicco Composition VTEX Typesetting Services Cover Printer Phoenix Color Interior Printer The MapleVail Book Manufacturing Group Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobalds Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright 2005, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or me- chanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elseviers Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting Customer Support and then Obtaining Permissions. Library of Congress Cataloging-in-Publication Data Appication submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-059876-0 Case bound ISBN: 0-12-088584-0 International Students Edition For all information on all Elsevier Academic Press Publications visit our Web site at www.books.elsevier.com Printed in the United States of America 05 06 07 08 09 10 9 8 7 6 5 4 3 2 1 6. CONTENTS Preface xi 1 Vector Analysis 1 1.1 Denitions, Elementary Approach . . . . . . . . . . . . . . . . . . . . . 1 1.2 Rotation of the Coordinate Axes . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Scalar or Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Vector or Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Triple Scalar Product, Triple Vector Product . . . . . . . . . . . . . . . 25 1.6 Gradient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7 Divergence, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.8 Curl, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.9 Successive Applications of . . . . . . . . . . . . . . . . . . . . . . . 49 1.10 Vector Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.11 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.12 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.13 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1.14 Gauss Law, Poissons Equation . . . . . . . . . . . . . . . . . . . . . . 79 1.15 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.16 Helmholtzs Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2 Vector Analysis in Curved Coordinates and Tensors 103 2.1 Orthogonal Coordinates in R3 . . . . . . . . . . . . . . . . . . . . . . . 103 2.2 Differential Vector Operators . . . . . . . . . . . . . . . . . . . . . . . 110 2.3 Special Coordinate Systems: Introduction . . . . . . . . . . . . . . . . 114 2.4 Circular Cylinder Coordinates . . . . . . . . . . . . . . . . . . . . . . . 115 2.5 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 123 v 7. vi Contents 2.6 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.7 Contraction, Direct Product . . . . . . . . . . . . . . . . . . . . . . . . 139 2.8 Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2.9 Pseudotensors, Dual Tensors . . . . . . . . . . . . . . . . . . . . . . . 142 2.10 General Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.11 Tensor Derivative Operators . . . . . . . . . . . . . . . . . . . . . . . . 160 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3 Determinants and Matrices 165 3.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.3 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.4 Hermitian Matrices, Unitary Matrices . . . . . . . . . . . . . . . . . . 208 3.5 Diagonalization of Matrices . . . . . . . . . . . . . . . . . . . . . . . . 215 3.6 Normal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4 Group Theory 241 4.1 Introduction to Group Theory . . . . . . . . . . . . . . . . . . . . . . . 241 4.2 Generators of Continuous Groups . . . . . . . . . . . . . . . . . . . . . 246 4.3 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 261 4.4 Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . 266 4.5 Homogeneous Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . 278 4.6 Lorentz Covariance of Maxwells Equations . . . . . . . . . . . . . . . 283 4.7 Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.8 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 5 Innite Series 321 5.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 5.2 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 5.3 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 5.4 Algebra of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 5.5 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5.6 Taylors Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 5.7 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.8 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 5.9 Bernoulli Numbers, EulerMaclaurin Formula . . . . . . . . . . . . . . 376 5.10 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.11 Innite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 6 Functions of a Complex Variable I Analytic Properties, Mapping 403 6.1 Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.2 CauchyRiemann Conditions . . . . . . . . . . . . . . . . . . . . . . . 413 6.3 Cauchys Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 418 8. Contents vii 6.4 Cauchys Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.5 Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 6.6 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 6.7 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 6.8 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 7 Functions of a Complex Variable II 455 7.1 Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 7.2 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 7.3 Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . . 489 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8 The Gamma Function (Factorial Function) 499 8.1 Denitions, Simple Properties . . . . . . . . . . . . . . . . . . . . . . . 499 8.2 Digamma and Polygamma Functions . . . . . . . . . . . . . . . . . . . 510 8.3 Stirlings Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 8.4 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 8.5 Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . 527 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 9 Differential Equations 535 9.1 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 535 9.2 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . 543 9.3 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 9.4 Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 9.5 Series SolutionsFrobenius Method . . . . . . . . . . . . . . . . . . . 565 9.6 A Second Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 9.7 Nonhomogeneous EquationGreens Function . . . . . . . . . . . . . 592 9.8 Heat Flow, or Diffusion, PDE . . . . . . . . . . . . . . . . . . . . . . . 611 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 10 SturmLiouville TheoryOrthogonal Functions 621 10.1 Self-Adjoint ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 10.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 10.3 GramSchmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . 642 10.4 Completeness of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 649 10.5 Greens FunctionEigenfunction Expansion . . . . . . . . . . . . . . . 662 Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 11 Bessel F