Mathematical Methods for Physics and Engineering - Matematica.NET

CONTENTS12.2 The Fourier coefficients 41712.3 Symmetry considerations 41912.4 Discontinuous functions 42012.5 Non-periodic functions 42212.6 Integration and differentiation 42412.7 Complex Fourier series 42412.8 Parseval’s theorem 42612.9 Exercises 42712.10 Hints and answers 43113 Integral transforms 43313.1 Fourier transforms 433The uncertainty principle; Fraunhofer diffraction; the Dirac δ-function;relation of the δ-function to Fourier transforms; properties of Fouriertransforms; odd and even functions; convolution and deconvolution; correlationfunctions and energy spectra; Parseval’s theorem; Fourier transforms in higherdimensions13.2 Laplace transforms 453Laplace transforms of derivatives and integrals; other properties of Laplacetransforms13.3 Concluding remarks 45913.4 Exercises 46013.5 Hints and answers 46614 First-order ordinary differential equations 46814.1 General form of solution 46914.2 First-degree first-order equations 470Separable-variable equations; exact equations; inexact equations, integratingfactors; linear equations; homogeneous equations; isobaric equations;Bernoulli’s equation; miscellaneous equations14.3 Higher-degree first-order equations 480Equations soluble for p; forx; fory; Clairaut’s equation14.4 Exercises 48414.5 Hints and answers 48815 Higher-order ordinary differential equations 49015.1 Linear equations with constant coefficients 492Finding the complementary function y c (x); finding the particular integraly p (x); constructing the general solution y c (x) +y p (x); linear recurrencerelations; Laplace transform method15.2 Linear equations with variable coefficients 503The Legendre and Euler linear equations; exact equations; partially knowncomplementary function; variation of parameters; Green’s functions; canonicalform for second-order equationsx