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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