Preface | p. xiii |
Vector Analysis | p. 1 |
Definitions, Elementary Approach | p. 1 |
Rotation of the Coordinate Axes | p. 8 |
Scalar or Dot Product | p. 13 |
Vector or Cross Product | p. 19 |
Triple Scalar Product, Triple Vector Product | p. 27 |
Gradient, [down triangle, open] | p. 35 |
Divergence, [down triangle, open] | p. 40 |
Curl, [down triangle, open] x | p. 44 |
Successive Applications of [down triangle, open] | p. 51 |
Vector Integration | p. 55 |
Gauss's Theorem | p. 61 |
Stokes's Theorem | p. 65 |
Potential Theory | p. 69 |
Gauss's Law, Poisson's Equation | p. 80 |
Dirac Delta Function | p. 84 |
Helmholtz's Theorem | p. 96 |
Curved Coordinates, Tensors | p. 103 |
Orthogonal Coordinates | p. 103 |
Differential Vector Operators | p. 108 |
Special Coordinate Systems: Introduction | p. 113 |
Circular Cylindrical Coordinates | p. 114 |
Spherical Polar Coordinates | p. 121 |
Tensor Analysis | p. 131 |
Contraction, Direct Product | p. 137 |
Quotient Rule | p. 139 |
Pseudotensors, Dual Tensors | p. 141 |
Non-Cartesian Tensors | p. 150 |
Tensor Derivative Operators | p. 160 |
Determinants and Matrices | p. 165 |
Determinants | p. 165 |
Matrices | p. 174 |
Orthogonal Matrices | p. 192 |
Hermitian Matrices, Unitary Matrices | p. 206 |
Diagonalization of Matrices | p. 213 |
Normal Matrices | p. 227 |
Group Theory | p. 237 |
Introduction to Group Theory | p. 237 |
Generators of Continuous Groups | p. 242 |
Orbital Angular Momentum | p. 258 |
Angular Momentum Coupling | p. 263 |
Homogeneous Lorentz Group | p. 275 |
Lorentz Covariance of Maxwell's Equations | p. 278 |
Discrete Groups | p. 286 |
Infinite Series | p. 303 |
Fundamental Concepts | p. 303 |
Convergence Tests | p. 306 |
Alternating Series | p. 322 |
Algebra of Series | p. 325 |
Series of Functions | p. 329 |
Taylor's Expansion | p. 334 |
Power Series | p. 346 |
Elliptic Integrals | p. 354 |
Bernoulli Numbers, Euler-Maclaurin Formula | p. 360 |
Asymptotic Series | p. 373 |
Infinite Products | p. 381 |
Functions of a Complex Variable I | p. 389 |
Complex Algebra | p. 390 |
Cauchy-Riemann Conditions | p. 399 |
Cauchy's Integral Theorem | p. 404 |
Cauchy's Integral Formula | p. 411 |
Laurent Expansion | p. 416 |
Mapping | p. 425 |
Conformal Mapping | p. 434 |
Functions of a Complex Variable II | p. 439 |
Singularities | p. 439 |
Calculus of Residues | p. 444 |
Dispersion Relations | p. 469 |
Method of Steepest Descents | p. 477 |
Differential Equations | p. 487 |
Partial Differential Equations | p. 487 |
First-Order Differential Equations | p. 496 |
Separation of Variables | p. 506 |
Singular Points | p. 516 |
Series Solutions--Frobenius's Method | p. 518 |
A Second Solution | p. 533 |
Nonhomogeneous Equation--Green's Function | p. 548 |
Numerical Solutions | p. 567 |
Sturm-Liouville Theory | p. 575 |
Self-Adjoint ODEs | p. 575 |
Hermitian Operators | p. 588 |
Gram-Schmidt Orthogonalization | p. 596 |
Completeness of Eigenfunctions | p. 604 |
Green's Function--Eigenfunction Expansion | p. 616 |
Gamma-Factorial Function | p. 631 |
Definitions, Simple Properties | p. 631 |
Digamma and Polygamma Functions | p. 643 |
Stirling's Series | p. 649 |
The Beta Function | p. 654 |
Incomplete Gamma Function | p. 660 |
Bessel Functions | p. 669 |
Bessel Functions of the First Kind J[subscript v](x) | p. 669 |
Orthogonality | p. 688 |
Neumann Functions, Bessel Functions of the Second Kind | p. 694 |
Hankel Functions | p. 702 |
Modified Bessel Functions I[subscript v](x) and K[subscript v](x) | p. 709 |
Asymptotic Expansions | p. 716 |
Spherical Bessel Functions | p. 722 |
Legendre Functions | p. 739 |
Generating Function | p. 739 |
Recurrence Relations | p. 748 |
Orthogonality | p. 755 |
Alternate Definitions | p. 767 |
Associated Legendre Functions | p. 771 |
Spherical Harmonics | p. 786 |
Orbital Angular Momentum Operators | p. 792 |
The Addition Theorem for Spherical Harmonics | p. 796 |
Integrals of Three Ys | p. 802 |
Legendre Functions of the Second Kind | p. 806 |
Vector Spherical Harmonics | p. 813 |
Special Functions | p. 817 |
Hermite Functions | p. 817 |
Laguerre Functions | p. 828 |
Chebyshev Polynomials | p. 839 |
Hypergeometric Functions | p. 850 |
Confluent Hypergeometric Functions | p. 855 |
Fourier Series | p. 863 |
General Properties | p. 863 |
Advantages, Uses of Fouries Series | p. 870 |
Applications of Fourier Series | p. 874 |
Properties of Fourier Series | p. 886 |
Gibbs Phenomenon | p. 893 |
Discrete Fourier Transform | p. 898 |
Integral Transforms | p. 905 |
Integral Transforms | p. 905 |
Development of the Fourier Integral | p. 909 |
Fourier Transforms--Inversion Theorem | p. 911 |
Fourier Transform of Derivatives | p. 920 |
Convolution Theorem | p. 924 |
Momentum Representation | p. 928 |
Transfer Functions | p. 935 |
Laplace Transforms | p. 938 |
Laplace Transform of Derivatives | p. 946 |
Other Properties | p. 953 |
Convolution or Faltungs Theorem | p. 965 |
Inverse Laplace Transform | p. 969 |
Integral Equations | p. 983 |
Introduction | p. 983 |
Integral Transforms, Generating Functions | p. 991 |
Neumann Series, Separable Kernels | p. 997 |
Hilbert-Schmidt Theory | p. 1009 |
Calculus of Variations | p. 1017 |
A Dependent and an Independent Variable | p. 1018 |
Applications of the Euler Equation | p. 1023 |
Several Dependent Variables | p. 1031 |
Several Independent Variables | p. 1036 |
Several Dependent and Independent Variables | p. 1038 |
Lagrangian Multipliers | p. 1039 |
Variation With Constraints | p. 1045 |
Rayleigh-Ritz Variational Technique | p. 1052 |
Nonlinear Methods and Chaos | p. 1059 |
Introduction | p. 1059 |
The Logistic Map | p. 1060 |
Sensitivity to Initial Conditions | p. 1064 |
Nonlinear Differential Equations | p. 1068 |
Real Zeros of a Function | p. 1085 |
Gaussian Quadrature | p. 1089 |
Index | p. 1097 |
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