Mathematical Methods for Physicists, 6th Edition, Arfken & Weber PDF

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Summary

V e ~ d 3 r = / B d a , (Gauss), L ( ~ x A ) . d a = A-dl, (Stokes) S (@v2y?- y?v2@)d3r (@V@- y?V@) da, (Green) Cumed Orthogonal Coordinates 3 I,'~jlinrlerCoordirtcr,les !%frttiemriticcrlChlzslcrntu e = 2.718281828, sr = 3.14159265, In 10 = 2.302585093, 1 rad = 57.29577951°, lo= 0.0174532925 ra...

Description

V e ~ d 3 r = / B d a , (Gauss),

L ( ~ x A ) . d a = A-dl,

(Stokes)

S

(@v2y?- y?v2@)d3r

(@V@- y?V@) da,

(Green)

3

Cumed Orthogonal Coordinates I,'~jlinrlerCoordirtcr,les

!%frttiemriticcrlChlzslcrntu

e = 2.718281828, sr = 3.14159265, In 10 = 2.302585093, 1 rad = 57.29577951°, lo= 0.0174532925 rad,

(Euler-Mascheroni number) 1 Bl = -2

1 1 1 &=g,B'=&=-&=30 ' 42'

(Bernoulli numbers) "'

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

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MATHEMATICAL METHODS FOR PHYSICISTS SIXTH EDITION

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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 mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. 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

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CONTENTS

Preface 1

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Vector Analysis 1.1 Definitions, Elementary Approach . . . . . . 1.2 Rotation of the Coordinate Axes . . . . . . . 1.3 Scalar or Dot Product . . . . . . . . . . . . 1.4 Vector or Cross Product . . . . . . . . . . . 1.5 Triple Scalar Product, Triple Vector Product 1.6 Gradient, ∇ . . . . . . . . . . . . . . . . . . 1.7 Divergence, ∇ . . . . . . . . . . . . . . . . . 1.8 Curl, ∇× . . . . . . . . . . . . . . . . . . . 1.9 Successive Applications of ∇ . . . . . . . . 1.10 Vector Integration . . . . . . . . . . . . . . . 1.11 Gauss’ Theorem . . . . . . . . . . . . . . . . 1.12 Stokes’ Theorem . . . . . . . . . . . . . . . 1.13 Potential Theory . . . . . . . . . . . . . . . 1.14 Gauss’ Law, Poisson’s Equation . . . . . . . 1.15 Dirac Delta Function . . . . . . . . . . . . . 1.16 Helmholtz’s Theorem . . . . . . . . . . . . . Additional Readings . . . . . . . . . . . . .

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1 1 7 12 18 25 32 38 43 49 54 60 64 68 79 83 95 101

Vector Analysis in Curved Coordinates and Tensors 2.1 Orthogonal Coordinates in R3 . . . . . . . . . . 2.2 Differential Vector Operators . . . . . . . . . . 2.3 Special Coordinate Systems: Introduction . . . 2.4 Circular Cylinder Coordinates . . . . . . . . . . 2.5 Spherical Polar Coordinates . . . . . . . . . . .

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103 103 110 114 115 123

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Contents 2.6 2.7 2.8 2.9 2.10 2.11

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Tensor Analysis . . . . . . . . Contraction, Direct Product . Quotient Rule . . . . . . . . . Pseudotensors, Dual Tensors General Tensors . . . . . . . . Tensor Derivative Operators . Additional Readings . . . . .

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133 139 141 142 151 160 163

Determinants and Matrices 3.1 Determinants . . . . . . . . . . . . . . 3.2 Matrices . . . . . . . . . . . . . . . . . 3.3 Orthogonal Matrices . . . . . . . . . . 3.4 Hermitian Matrices, Unitary Matrices 3.5 Diagonalization of Matrices . . . . . . 3.6 Normal Matrices . . . . . . . . . . . . Additional Readings . . . . . . . . . .

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165 165 176 195 208 215 231 239

Group Theory 4.1 Introduction to Group Theory . . . . . . . . 4.2 Generators of Continuous Groups . . . . . . 4.3 Orbital Angular Momentum . . . . . . . . . 4.4 Angular Momentum Coupling . . . . . . . . 4.5 Homogeneous Lorentz Group . . . . . . . . 4.6 Lorentz Covariance of Maxwell’s Equations 4.7 Discrete Groups . . . . . . . . . . . . . . . . 4.8 Differential Forms . . . . . . . . . . . . . . Additional Readings . . . . . . . . . . . . .

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241 241 246 261 266 278 283 291 304 319

Infinite Series 5.1 Fundamental Concepts . . . . . . . . . . . . . 5.2 Convergence Tests . . . . . . . . . . . . . . . 5.3 Alternating Series . . . . . . . . . . . . . . . . 5.4 Algebra of Series . . . . . . . . . . . . . . . . 5.5 Series of Functions . . . . . . . . . . . . . . . 5.6 Taylor’s Expansion . . . . . . . . . . . . . . . 5.7 Power Series . . . . . . . . . . . . . . . . . . 5.8 Elliptic Integrals . . . . . . . . . . . . . . . . 5.9 Bernoulli Numbers, Euler–Maclaurin Formula 5.10 Asymptotic Series . . . . . . . . . . . . . . . . 5.11 Infinite Products . . . . . . . . . . . . . . . . Additional Readings . . . . . . . . . . . . . .

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321 321 325 339 342 348 352 363 370 376 389 396 401

Functions of a Complex Variable I Analytic Properties, Mapping 6.1 Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Cauchy–Riemann Conditions . . . . . . . . . . . . . . . . . . . . . . . 6.3 Cauchy’s Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

403 404 413 418

Contents 6.4 6.5 6.6 6.7 6.8

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Cauchy’s Integral Formula Laurent Expansion . . . . Singularities . . . . . . . . Mapping . . . . . . . . . . Conformal Mapping . . . Additional Readings . . .

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425 430 438 443 451 453

Functions of a Complex Variable II 7.1 Calculus of Residues . . . . . 7.2 Dispersion Relations . . . . . 7.3 Method of Steepest Descents . Additional Readings . . . . .

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455 455 482 489 497

The Gamma Function (Factorial Function) 8.1 Definitions, Simple Properties . . . . . 8.2 Digamma and Polygamma Functions . 8.3 Stirling’s Series . . . . . . . . . . . . . 8.4 The Beta Function . . . . . . . . . . . 8.5 Incomplete Gamma Function . . . . . Additional Readings . . . . . . . . . .

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499 499 510 516 520 527 533

Differential Equations 9.1 Partial Differential Equations . . . . . . . . . . 9.2 First-Order Differential Equations . . . . . . . 9.3 Separation of Variables . . . . . . . . . . . . . . 9.4 Singular Points . . . . . . . . . . . . . . . . . . 9.5 Series Solutions—Frobenius’ Method . . . . . . 9.6 A Second Solution . . . . . . . . . . . . . . . . . 9.7 Nonhomogeneous Equation—Green’s Function 9.8 Heat Flow, or Diffusion, PDE . . . . . . . . . . Additional Readings . . . . . . . . . . . . . . .

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535 535 543 554 562 565 578 592 611 618

10 Sturm–Liouville Theory—Orthogonal Functions 10.1 Self-Adjoint ODEs . . . . . . . . . . . . . . 10.2 Hermitian Operators . . . . . . . . . . . . . 10.3 Gram–Schmidt Orthogonalization . . . . . . 10.4 Completeness of Eigenfunctions . . . . . . . 10.5 Green’s Function—Eigenfunction Expansion Additional Readings . . . . . . . . . . . . .

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621 622 634 642 649 662 674

11 Bessel Functions 11.1 Bessel Functions of the First Kind, Jν (x) . . 11.2 Orthogonality . . . . . . . . . . . . . . . . . 11.3 Neumann Functions . . . . . . . . . . . . . 11.4 Hankel Functions . . . . . . . . . . . . . . . 11.5 Modified Bessel Functions, Iν (x) and Kν (x)

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675 675 694 699 707 713

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Contents 11.6 11.7

Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Legendre Functions 12.1 Generating Function . . . . . . . . . . . . . 12.2 Recurrence Relations . . . . . . . . . . . . . 12.3 Orthogonality . . . . . . . . . . . . . . . . . 12.4 Alternate Definitions . . . . . . . . . . . . . 12.5 Associated Legendre Functions . . . . . . . 12.6 Spherical Harmonics . . . . . . . . . . . . . 12.7 Orbital Angular Momentum Operators . . . 12.8 Addition Theorem for Spherical Harmonics 12.9 Integrals of Three Y’s . . . . . . . . . . . . . 12.10 Legendre Functions of the Second Kind . . . 12.11 Vector Spherical Harmonics . . . . . . . . . Additional Readings . . . . . . . . . . . . .

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741 741 749 756 767 771 786 793 797 803 806 813 816

13 More Special Functions 13.1 Hermite Functions . . . . . . . . . . 13.2 Laguerre Functions . . . . . . . . . . 13.3 Chebyshev Polynomials . . . . . . . 13.4 Hypergeometric Functions . . . . . . 13.5 Confluent Hypergeometric Functions 13.6 Mathieu Functions . . . . . . . . . . Additional Readings . . . . . . . . .

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817 817 837 848 859 863 869 879

14 Fourier Series 14.1 General Properties . . . . . . . . . . . . . 14.2 Advantages, Uses of Fourier Series . . . . 14.3 Applications of Fourier Series . . . . . . . 14.4 Properties of Fourier Series . . . . . . . . 14.5 Gibbs Phenomenon . . . . . . . . . . . . . 14.6 Discrete Fourier Transform . . . . . . . . 14.7 Fourier Expansions of Mathieu Functions Additional Readings . . . . . . . . . . . .

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881 881 888 892 903 910 914 919 929

15 Integral Transforms 15.1 Integral Transforms . . . . . . . . . . . . 15.2 Development of the Fourier Integral . . . 15.3 Fourier Transforms—Inversion Theorem 15.4 Fourier Transform of Derivatives . . . . 15.5 Convolution Theorem . . . . . . . . . . . 15.6 Momentum Representation . . . . . . . . 15.7 Transfer Functions . . . . . . . . . . . . 15.8 Laplace Transforms . . . . . . . . . . . .

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931 931 936 938 946 951 955 961 965

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Contents 15.9 15.10 15.11 15.12

Laplace Transform of Derivatives Other Properties . . . . . . . . . Convolution (Faltungs) Theorem Inverse Laplace Transform . . . . Additional Readings . . . . . . .

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16 Integral Equations 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . 16.2 Integral Transforms, Generating Functions . . . . 16.3 Neumann Series, Separable (Degenerate) Kernels 16.4 Hilbert–Schmidt Theory . . . . . . . . . . . . . . Additional Readings . . . . . . . . . . . . . . . .

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1005 1005 1012 1018 1029 1036

17 Calculus of Variations 17.1 A Dependent and an Independent Variable . . 17.2 Applications of the Euler Equation . . . . . . 17.3 Several Dependent Variables . . . . . . . . . . 17.4 Several Independent Variables . . . . . . . . . 17.5 Several Dependent and Independent Variables 17.6 Lagrangian Multipliers . . . . . . . ...