Mathematical Methods for Physicists, Fifth Edition

1 Vector Analysis 1 -- 1.2 Rotation Of The Coordinate Axes 8 -- 1.3 Scalar Or Dot Product 13 -- 1.4 Vector Or Cross Product 19 -- 1.5 Triple Scalar Product, Triple Vector Product 27 -- 1.6 Gradient, [down Triangle, Open] 35 -- 1.7 Divergence, [down Triangle, Open] 40 -- 1.8 Curl, [down Triangle, Open] X 44 -- 1.9 Successive Applications Of [down Triangle, Open] 51 -- 1.10 Vector Integration 55 -- 1.11 Gauss's Theorem 61 -- 1.12 Stokes's Theorem 65 -- 1.13 Potential Theory 69 -- 1.14 Gauss's Law, Poisson's Equation 80 -- 1.15 Dirac Delta Function 84 -- 1.16 Helmholtz's Theorem 96 -- 2 Curved Coordinates, Tensors 103 -- 2.1 Orthogonal Coordinates 103 -- 2.2 Differential Vector Operators 108 -- 2.3 Special Coordinate Systems: Introduction 113 -- 2.4 Circular Cylindrical Coordinates 114 -- 2.5 Spherical Polar Coordinates 121 -- 2.6 Tensor Analysis 131 -- 2.7 Contraction, Direct Product 137 -- 2.8 Quotient Rule 139 -- 2.9 Pseudotensors, Dual Tensors 141 -- 2.10 Non-cartesian Tensors 150 -- 2.11 Tensor Derivative Operators 160 -- 3 Determinants And Matrices 165 -- 3.1 Determinants 165 -- 3.2 Matrices 174 -- 3.3 Orthogonal Matrices 192 -- 3.4 Hermitian Matrices, Unitary Matrices 206 -- 3.5 Diagonalization Of Matrices 213 -- 3.6 Normal Matrices 227 -- 4 Group Theory 237 -- 4.1 Introduction To Group Theory 237 -- 4.2 Generators Of Continuous Groups 242 -- 4.3 Orbital Angular Momentum 258 -- 4.4 Angular Momentum Coupling 263 -- 4.5 Homogeneous Lorentz Group 275 -- 4.6 Lorentz Covariance Of Maxwell's Equations 278 -- 4.7 Discrete Groups 286 -- 5 Infinite Series 303 -- 5.2 Convergence Tests 306 -- 5.3 Alternating Series 322 -- 5.4 Algebra Of Series 325 -- 5.5 Series Of Functions 329 -- 5.6 Taylor's Expansion 334 -- 5.7 Power Series 346 -- 5.8 Elliptic Integrals 354 -- 5.9 Bernoulli Numbers, Euler-maclaurin Formula 360 -- 5.10 Asymptotic Series 373 -- 5.11 Infinite Products 381 -- 6 Functions Of A Complex Variable I 389 -- 6.1 Complex Algebra 390 -- 6.2 Cauchy-riemann Conditions 399 -- 6.3 Cauchy's Integral Theorem 404 -- 6.4 Cauchy's Integral Formula 411 -- 6.5 Laurent Expansion 416 -- 6.6 Mapping 425 -- 6.7 Conformal Mapping 434 -- 7 Functions Of A Complex Variable Ii 439 -- 7.1 Singularities 439 -- 7.2 Calculus Of Residues 444 -- 7.3 Dispersion Relations 469 -- 7.4 Method Of Steepest Descents 477 -- 8 Differential Equations 487 -- 8.1 Partial Differential Equations 487 -- 8.2 First-order Differential Equations 496 -- 8.3 Separation Of Variables 506 -- 8.4 Singular Points 516 -- 8.5 Series Solutions--frobenius's Method 518 -- 8.6 A Second Solution 533 -- 8.7 Nonhomogeneous Equation--green's Function 548 -- 8.8 Numerical Solutions 567 -- 9 Sturm-liouville Theory 575 -- 9.1 Self-adjoint Odes 575 -- 9.2 Hermitian Operators 588 -- 9.3 Gram-schmidt Orthogonalization 596 -- 9.4 Completeness Of Eigenfunctions 604 -- 9.5 Green's Function--eigenfunction Expansion 616 -- 10 Gamma-factorial Function 631 -- 10.1 Definitions, Simple Properties 631 -- 10.2 Digamma And Polygamma Functions 643 -- 10.3 Stirling's Series 649 -- 10.4 Beta Function 654 -- 10.5 Incomplete Gamma Function 660 -- 11 Bessel Functions 669 -- 11.1 Bessel Functions Of The First Kind J[subscript V](x) 669 -- 11.2 Orthogonality 688 -- 11.3 Neumann Functions, Bessel Functions Of The Second Kind 694 -- 11.4 Hankel Functions 702 -- 11.5 Modified Bessel Functions I[subscript V](x) And K[subscript V](x) 709 -- 11.6 Asymptotic Expansions 716 -- 11.7 Spherical Bessel Functions 722 -- 12 Legendre Functions 739 -- 12.1 Generating Function 739 -- 12.2 Recurrence Relations 748 -- 12.3 Orthogonality 755 -- 12.4 Alternate Definitions 767 -- 12.5 Associated Legendre Functions 771 -- 12.6g Spherical Harmonics 786 -- 12.7 Orbital Angular Momentum Operators 792 -- 12.8 Addition Theorem For Spherical Harmonics 796 -- 12.9 Integrals Of Three Ys 802 -- 12.10 Legendre Functions Of The Second Kind 806 -- 12.11 Vector Spherical Harmonics 813 -- 13 Special Functions 817 -- 13.1 Hermite Functions 817 -- 13.2 Laguerre Functions 828 -- 13.3 Chebyshev Polynomials 839 -- 13.4 Hypergeometric Functions 850 -- 13.5 Confluent Hypergeometric Functions 855 -- 14 Fourier Series 863 -- 14.1 General Properties 863 -- 14.2 Advantages, Uses Of Fouries Series 870 -- 14.3 Applications Of Fourier Series 874 -- 14.4 Properties Of Fourier Series 886 -- 14.5 Gibbs Phenomenon 893 -- 14.6 Discrete Fourier Transform 898 -- 15 Integral Transforms 905 -- 15.1 Integral Transforms 905 -- 15.2 Development Of The Fourier Integral 909 -- 15.3 Fourier Transforms--inversion Theorem 911 -- 15.4 Fourier Transform Of Derivatives 920 -- 15.5 Convolution Theorem 924 -- 15.6 Momentum Representation 928 -- 15.7 Transfer Functions 935 -- 15.8 Laplace Transforms 938 -- 15.9 Laplace Transform Of Derivatives 946 -- 15.10 Other Properties 953 -- 15.11 Convolution Or Faltungs Theorem 965 -- 15.12 Inverse Laplace Transform 969 -- 16 Integral Equations 983 -- 16.2 Integral Transforms, Generating Functions 991 -- 16.3 Neumann Series, Separable Kernels 997 -- 16.4 Hilbert-schmidt Theory 1009 -- 17 Calculus Of Variations 1017 -- 17.1 A Dependent And An Independent Variable 1018 -- 17.2 Applications Of The Euler Equation 1023 -- 17.3 Several Dependent Variables 1031 -- 17.4 Several Independent Variables 1036 -- 17.5 Several Dependent And Independent Variables 1038 -- 17.6 Lagrangian Multipliers 1039 -- 17.7 Variation With Constraints 1045 -- 17.8 Rayleigh-ritz Variational Technique 1052 -- 18 Nonlinear Methods And Chaos 1059 -- 18.2 Logistic Map 1060 -- 18.3 Sensitivity To Initial Conditions 1064 -- 18.4 Nonlinear Differential Equations 1068 -- Appendix 1 Real Zeros Of A Function 1085 -- Appendix 2 Gaussian Quadrature 1089. George B. Arfken, Hans J. Weber. Includes Bibliographical References And Index.