Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXdiscontinuous functions, 420–422error term, 430integration, 424non-periodic functions, 422–424orthogonality of terms, 417complex case, 425Parseval’s theorem, 426standard form, 417summation of series, 427symmetry considerations, 419uses, 415Fourier series, examplessquare-wave, 418x, 424, 425x 2 , 422x 3 , 424Fourier sine transforms, 445Fourier transforms, 433–453as generalisation of Fourier series, 433–435convolution, 446–449and the Dirac δ-function, 447associativity, commutativity, distributivity,447definition, 447resolution function, 446convolution theorem, 448correlation functions, 449–451cosine transforms, 446deconvolution, 449definition, 435discrete, 462evaluation using convolution theorem, 448for integral equations, 809–812for PDE, 749–751Fourier-related (conjugate) variables, 436in higher dimensions, 451–453inverse, definition, 435odd and even functions, 445Parseval’s theorem, 450properties: differentiation, exponentialmultiplication, integration, scaling,translation, 444relation to Dirac δ-function, 442sine transforms, 445Fourier transforms, examplesconvolution, 448damped harmonic oscillator, 451Dirac δ-function, 443exponential decay function, 435Gaussian (normal) distribution, 435rectangular distribution, 442spherically symmetric functions, 452two narrow slits, 448two wide slits, 438, 448Fourier’s inversion theorem, 435Fraunhofer diffraction, 437–439diffraction grating, 461two narrow slits, 448two wide slits, 438, 448Fredholm integral equations, 805eigenvalues, 808operator form, 806with separable kernel, 807Fredholm theory, 815Frenet–Serret formulae, 343Fresnel integrals, 913Frobenius series, 539Fuch’s theorem, 539function of a matrix, 255functional, 776functions of a complex variable, 825–839,853–858analyticity, 826behaviour at infinity, 839branch points, 835Cauchy integrals, 851–853Cauchy–Riemann relations, 827–830conformal transformations, 839–844derivative, 825differentiation, 825–830identity theorem, 854Laplace equation, 829, 871Laurent expansion, 855–858multivalued and branch cuts, 835–837, 885particular functions, 832–835poles, 837power series, 830–832real and imaginary parts, 825, 830singularities, 826, 837–839Taylor expansion, 853–855zeros, 839, 879–882functions of one real variabledecomposition into even and odd functions,416differentiation of, 41–50Fourier series, see Fourier seriesintegration of, 59–72limits, see limitsmaxima and minima of, 50–52stationary values of, 50–52Taylor series, see Taylor seriesfunctions of several real variableschain rule, 157differentiation of, 151–179integration of, see multiple integrals,evaluationmaxima and minima, 162–167points of inflection, 162–167rates of change, 153–155saddle points, 162–167stationary values, 162–167Taylor series, 160–162fundamental solution, 757fundamental theorem ofalgebra, 83, 85, 868calculus, 61complex numbers, see de Moivre’s theoremgamma distribution, 1153, 1191gamma function1315