mathematical methods for physicists - gbv
TRANSCRIPT
MATHEMATICAL METHODS FOR PHYSICISTS Fourth Edition
George B. Arfken Miami University Oxford, Ohio
Hans J. Weber University of Virginia Charlottesville, Virginia
Academic Press San Diego New York Boston London Sydney Tokyo Toronto
-.,.
CONTENTS
Preface xv Introduction xvii
1 VECTOR ANALYSIS
1.1 Definitions, Elementary Approach 1
1.2 Rotation of the Coordinate Axes 7
1.3 Scalar or Dot Product 13
1.4 Vector or Cross Product 18
1.5 Triple Scalar Product, Triple Vector Product 26
1.6 Gradient, V 33
1.7 Divergence, V - 38
1.8 C u r l , V x 42
1.9 Successive Applications of V 48
1.10 Vector Integration 52
1.11 Gauss's Theorem 58
1.12 Stokes's Theorem 62
1.13 Potential Theory 66
1.14 Gauss's Law, Poisson's Equation 77
1.15 Dirac Delta Function 81
1.16 Helmholtz's Theorem 92
vii
v.
•
viii Contents
2 VECTOR ANALYSIS IN CURYED COORDINATES AND TENSORS
2.1 Orthogonal Coordinates 100
2.2 Differential Vector Operators 104
2.3 Special Coordinate Systems: Introduction 109
2.4 Circular Cylindrical Coordinates (p , cp, z) 110
2.5 Spherical Polar Coordinates (r, 0 , cp) 117
2.6 Tensor Analysis 126
2.7 Contraction, Direct Product 132
2.8 Quotient Rule 134
2.9 Pseudotensors, Dual Tensors 135
2.10 Noncartesian Tensors, Covariant Differentiation 145
2.11 Tensor Differential Operators 152
3 DETERMINANTS AND MATRICES
3.1 Determinants 156
3.2 Matrices 165
3.3 Orthogonal Matrices 181
3.4 Hermitian Matrices, Unitary Matrices 194
3.5 Diagonalization of Matrices 201
3.6 Normal Matrices 213
4 GROUP THEORY
4.1 Introduction to Group Theory 223
4.2 Generators of Continuous Groups 227
4.3 Orbital Angular Momentum 243
4.4 Angular Momentum Coupling 247
4.5 Homogeneous Lorentz Group 258
4.6 Lorentz Covariance of Maxwell's Equations 262
4.7 Discrete Groups 269
Contents ix
5
6
7
INFINITE SERIES
5.1 Fundamental Concepts 284
5.2 Convergence Tests 288
5.3 Alternating Series 302
5.4 Algebra of Series 304
5.5 Series of Functions 309
5.6 Taylor's Expansion 313
5.7 Power Series 324
5.8 Elliptic Integrals 331
5.9 Bernoulli Numbers, Euler-Maclaurin Formula 337
5.10 Asymptotic or Semiconvergent Series 349
5.11 Infinite Products 357
FUNCTIONS OF A COMPLEX VARIABLE 1: ANALYTIC PROPERTIES MAPPING
6.1 Complex Algebra 364
6.2 Cauchy—Riemann Conditions 372
6.3 Cauchy's Integral Theorem 377
6.4 Cauchy's Integral Formula 384
6.5 Laurent Expansion 389
6.6 Mapping 398
6.7 Conformal Mapping 406
FUNCTIONS OF A COMPLEX VARIABLE II: CALCULUS OF RESIDUES
7.1 Singularities 410
7.2 Calculus of Residues 414
7.3 Dispersion Relations 439
7.4 The Method of Steepest Descents 446
x Contents
8 1 DIFFERENTIAL EQUATIONS
8.1 Portial Differential Equations, Characteristics, and Boundary Conditions 456
8.2 First-Order Differential Equations 463
8.3 Separation of Variables 471
8.4 Singular Points 480
8.5 Series Solutions—Frobenius' Method 483
8.6 A Second Solution 497
8.7 Nonhomogeneous Equation—Green's Function 510
8.8 Numerical Solutions 529
9 STURM-LIOUVILLE THEORY— ORTHOGONAL FUNCTIONS
9.1 Self-Adjoint Differential Equations 537
9.2 Hermitian Operators 551
9.3 Gram-Schmidt Orthogonalization 558
9.4 Completeness of Eigenfunctions 565
9.5 Green's Function—Eigenfunction Expansion 577
10 THE GAMMA FUNCTION (FACTORIAL FUNCTION)
10.1 Definitions, Simple Properties 591
10.2 Digamma and Polygamma Functions 602
10.3 Stirling's Series 608
10.4 The Beta Function 613
10.5 The Incomplete Gamma Functions and Related Functions
Contents xi
11 BESSEL FUNCTIONS
11.1 Bessel Functions of the First Kind, Jv{x) 627
11.2 Orthogonality 645
11.3 Neumann Functions, Bessel Functions of the Second Kind, Wv(x) 651
11.4 Hankel Functions 658
11.5 Modified Bessel Functions, / v (x) and Kv(x) 664
11.6 Asymptotic Expansions 671
11.7 Spherical Bessel Functions 677
12 LEGENDRE FUNCTIONS
12.1 Generating Function 693
12.2 Recurrence Relations and Special Properties 701
12.3 Orthogonality 708
12.4 Alternate Definitions of Legendre Polynomials 719
12.5 Associated Legendre Functions 722
12.6 Spherical Harmonics 736
12.7 Orbital Angular Momentum Operators 742
12.8 The Addition Theorem for Spherical Harmonics 746
12.9 Integrals of the Product of Three Spherical Harmonics 751
12.10 Legendre Functions of the Second Kind, Qn(x) 755
12.11 Vector Spherical Harmonics 762
13 SPECIAL FUNCTIONS
13.1 Hermite Functions 766
13.2 Laguerre Functions 776
13.3 Chebyshev (Tschebyscheff) Polynomials 786
>.
xii Contents
13.4 Hypergeometric Functions 796
13.5 Confluent Hypergeometric Functions 801
14 FOURIER SERIES
14.1 General Properties 808
14.2 Advantages, Uses of Fourier Series 815
14.3 Applications of Fourier Series 818
14.4 Properties of Fourier Series 829
14.5 Gibbs Phenomenon 836
14.6 Discrete Orthogonality—Discrete Fourier Transform 840
15 INTEGRAL TRANSFORMS
15.1 Integral Transforms 846
15.2 Development of the Fourier Integral 850
15.3 Fourier Transforms—Inversion Theorem 852
15.4 Fourier Transform of Derivatives 860
15.5 Convolution Theorem 863
15.6 Momentum Representation 868
15.7 Transfer Functions 874
15.8 Elementary Laplace Transforms 877
15.9 Laplace Transform of Derivatives 885
15.10 Other Properties 892
15.11 Convolution or Faltung Theorem 904
15.12 Inverse Laplace Transformation 908
16 INTEGRAL EQUATIONS
16.1 Introduction 920
16.2 Integral Transforms, Generating Functions 927
16.3 Neumann Series, Separable (Degenerate) Kernels 933
16.4 Hilbert—Schmidt Theory 944
17 CALCULUS OF YARIATIONS
17.1 One-Dependent and One-Independent Variable 953
17.2 Applications of the Euler Equation 957
17.3 Generalizations, Several Dependent Variables 965
17.4 Several Independent Variables 970
17.5 More Than One Dependent, More Than One Independent Variable 972
17.6 Lagrangian Multipliers 973
17.7 Variation Subject to Constraints 978
17.8 Rayleigh-Ritz Variational Technique 986
18 NONLINEAR METHODS AND CHAOS
18.1 Introduction 992
18.2 The Logistic Map 993
18.3 Sensitivity to Initial Conditions and Parameters 997
18.4 Nonlinear Differential Equations 999
Appendix 1 Real Zeros of a Function 1005 Appendix 2 Gaussian Quadrature 1009 General References 1016 Index 1017