Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXas general factorial function, 636definition and properties, 636graph of, 637Gauss’s theorem, 765Gauss–Seidel iteration, 996–998Gaussian (normal) distribution N(µ, σ 2 ),1179–1189and binomial distribution, 1185and central limit theorem, 1195and Poisson distribution, 1187continuity correction, 1186CPF, 1018, 1181tabulation, 1182Fourier transform, 435integration with infinite limits, 202–204mean and variance, 1180–1184MGF, 1185, 1188multiple, 1188multivariate, 1209random number generation, 1018sigma limits, 1183standard variable, 1180Gaussian elimination with interchange, 995Gaussian integration, 1005–1009points and weights, 1008, 1010general tensorsalgebra, 938–941contraction, 939contravariant, 961covariant, 961dual, 949metric, 957–960physical applications, 957–960, 976pseudotensors, 964tensor densities, 964generalised likelihood ratio, 1282generating functionsassociated Laguerre polynomials, 623associated Legendre polynomials, 592Bessel functions, 613Chebyshev polynomials, 601Hermite polynomials, 627Laguerre polynomials, 620Legendre polynomials, 584–586generating functions in probability, 1157–1167,see also moment generating functions andprobability generating functionsgeodesics, 797, 976, 982geometric distribution, 1159, 1172geometric series, 117Gibbs’ free energy, 178Gibbs’ phenonmenon, 421gradient of a function ofone variable, 42several real variables, 153–155gradient of scalar, 348–352tensor form, 972gradient of vector, 937, 969gradient operator (grad), 348as integral, 398in curvilinear coordinates, 367in cylindrical polars, 360in spherical polars, 362tensor form, 972Gram–Schmidt orthogonalisation ofeigenfunctions of Hermitian operators, 562eigenvectors ofHermitian matrices, 277normal matrices, 275functions in a Hilbert space, 557gravitational fields and potentialsLaplace equation, 679Newton’s law, 339Poisson equation, 679, 744uniform disc, 771uniform ring, 742Green’s functions, 568–571, 751–767and boundary conditions, 512, 514and Dirac δ-function, 511and partial differential operators, 753and Wronskian, 527diffusion equation, 749Dirichlet problems, 756–765for ODE, 185, 511–516Neumann problems, 765–767particular integrals from, 514Poisson’s equation, 755Green’s theoremsapplications, 706, 754, 849in a plane, 384–387, 407in three dimensions, 402ground-state energyharmonic oscillator, 796hydrogen atom, 800group multiplication tables, 1050order three, 1062order four, 1050, 1052, 1061order five, 1062order six, 1055, 1061grouping terms as a test for convergence, 129groupsAbelian, 1044associative law, 1043cancellation law, 1046centre, 1069closure, 1043cyclic, 1061definition, 1043–1046direct product, 1072division axiom, 1046elements, 1043order, 1047finite, 1043identity element, 1043–1046inverse, 1043, 1046isomorphic, 1051mappings between, 1059–1061homomorphic, 1059–1061image, 1059isomorphic, 10591316