Analytic Methods in Physics

Charlie Harper, Prof., California State University, Hayward, CA, USA

VECTOR ANALYSIS The Cartesian Coordinate System Differentiation of Vector Functions Orthogonal Curvilinear Coordinates Problems Appedix I: SI Units Appendix II: Determinants MODERN ALGEBRAIC METHODS IN PHYSICS Matrix Analysis Essentials of Vector Spaces Essential Algebraic Structures Problems FUNCTIONS OF A COMPLEX VARIABLE Complex Variables and Their Representations The de Moivre Theorem Analytic Functions of a Complex Variable Contour Integrals The Taylor Series and Zeros of f(z) The Laurent Expansion Problems Appendix: Series CALCULUS OF RESIDUES Isolated Singular Points Evaluation of Residues The Cauchy Residue Theorem The Cauchy Principal Value Evaluation of Definite Integrals Dispersion Relations Conformal Transformations Multi Valued Functions Problems FOURIER SERIES The Fourier Cosine and Sine Series Change of Interval Complex Form of the Fourier Series Generalized Fourier Series and the Dirac Delta Function Summation of the Fourier Series The Gibbs Phenomenon Summary of Some Properties of Fourier Series Problems FOURIER TRANSFORMS Cosine and Sine Transforms The Transforms of Derivatives The Convolution Theorem Parseval s Relation Problems ORDINARY DIFFERENTIAL EQUATIONSFIRST ORDER LINEAR DIFFERENTIAL EQUATIONS The Bernoulli Differential Equation Second Order Linear Differential Equations Some Numerical Methods Problems PARTIAL DIFFERENTIAL EQUATIONS The Method of Separation of Variables Green s Functions in Potential Theory Some Numerical Methods Problems SPECIAL FUNCTIONS The Sturm Liouville Theory The Hermite Polynomials The Helmholtz Differential Equation in Spherical Coordinates and in Cylindrical Coordinates The Hypergeometric Function The Confluent Hypergeometric Function Other Special Functions Used in Physics Problems Worksheet 9.1: The Quantum Mechanical Linear Harmonic Oscillator Workshheet 9.2: The Legendre Differential Equation Worksheet 9.3: The Laguerre Differential Equation Workshheet 9.4: The Bessel Differential Equation Workshheet 9.5: The Hypergeometric Differential Equation INTEGRAL EQUATIONS Integral Equations with Separable Kernels and with Displacement Kernels The Neumann Series Method The Abel Problem Problems APPLIED FUNCTIONAL ANALYSIS Stationary Values of Certain Functions and Functionals Hamilton s Variational Principle in Mechanics Formulation of Hamiltonian Mechanics Continous Media and Fields Transitions to Quantum Mechanics Problems GEOMETRIC METHODS IN PHYSICS Transformation of Coordinates in Linear Spaces Contravariant and Covariant Tensors Tensor Algebra The Line Element Tensor Calculus Equation of the Geodesic Line Special Equations involving the Metric Tensor Exterior Differential Forms Problems References Index