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Mathematical Methods for Physicists by G.Arfken.pdf
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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Printed in the United States of America
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1
CONTENTS
Preface
1 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 ∇ . . . . . . . . . . . . . . . . . . . . . . .
Vector Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10
1.11 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12
1.13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Potential Theory
1.14 Gauss’ Law, Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . .
1.15 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.16 Helmholtz’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
2 Vector Analysis in Curved Coordinates and Tensors
2.1
2.2
2.3
2.4
2.5
Orthogonal Coordinates in R3 . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
Differential Vector Operators
Special Coordinate Systems: Introduction . . . . . . . . . . . . . . . .
Circular Cylinder Coordinates . . . . . . . . . . . . . . . . . . . . . . .
Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . .
xi
1
1
7
12
18
25
32
38
43
49
54
60
64
68
79
83
95
101
103
103
110
114
115
123
v
vi
Contents
Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
. . . . . . . . . . . . . . . . . . . . . . . .
2.7
Contraction, Direct Product
Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8
2.9
. . . . . . . . . . . . . . . . . . . . . . .
Pseudotensors, Dual Tensors
2.10 General Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tensor Derivative Operators . . . . . . . . . . . . . . . . . . . . . . . .
2.11
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
3 Determinants and Matrices
3.1
3.2
3.3
3.4
3.5
3.6
Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
Hermitian Matrices, Unitary Matrices
Diagonalization of Matrices . . . . . . . . . . . . . . . . . . . . . . . .
Normal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
4 Group Theory
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Introduction to Group Theory . . . . . . . . . . . . . . . . . . . . . . .
Generators of Continuous Groups . . . . . . . . . . . . . . . . . . . . .
Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . .
Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . .
Homogeneous Lorentz Group . . . . . . . . . . . . . . . . . . . . . . .
Lorentz Covariance of Maxwell’s Equations . . . . . . . . . . . . . . .
Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differential Forms
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
5
6
Infinite Series
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence Tests
Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Algebra of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Taylor’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power Series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elliptic Integrals
Bernoulli Numbers, Euler–Maclaurin Formula . . . . . . . . . . . . . .
Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Infinite Products
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
Functions of a Complex Variable I Analytic Properties, Mapping
6.1
6.2
6.3
Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
Cauchy–Riemann Conditions
Cauchy’s Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . .
133
139
141
142
151
160
163
165
165
176
195
208
215
231
239
241
241
246
261
266
278
283
291
304
319
321
321
325
339
342
348
352
363
370
376
389
396
401
403
404
413
418
Contents
6.4
6.5
6.6
6.7
6.8
Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . .
Laurent Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
7
Functions of a Complex Variable II
7.1
7.2
7.3
Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
8 The Gamma Function (Factorial Function)
8.1
8.2
8.3
8.4
8.5
Definitions, Simple Properties . . . . . . . . . . . . . . . . . . . . . . .
Digamma and Polygamma Functions . . . . . . . . . . . . . . . . . . .
Stirling’s Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
9 Differential Equations
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
First-Order Differential Equations
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . .
Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Series Solutions—Frobenius’ Method . . . . . . . . . . . . . . . . . . .
A Second Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonhomogeneous Equation—Green’s Function . . . . . . . . . . . . .
Heat Flow, or Diffusion, PDE . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
10 Sturm–Liouville Theory—Orthogonal Functions
Self-Adjoint ODEs
10.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Gram–Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . .
10.4 Completeness of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . .
10.5 Green’s Function—Eigenfunction Expansion . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings
11 Bessel Functions
Bessel Functions of the First Kind, Jν (x) . . . . . . . . . . . . . . . . .
11.1
11.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Neumann Functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Modified Bessel Functions, Iν (x) and Kν (x) . . . . . . . . . . . . . . .
vii
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430
438
443
451
453
455
455
482
489
497
499
499
510
516
520
527
533
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535
543
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713
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