Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXgeneral properties, see anti-Hermitianmatricesantisymmetric tensors, 938, 941antithetic variates, in Monte Carlo methods,1014aperture function, 437approximately equal ≈, definition, 132arbitrary parameters for ODE, 469arc length ofplane curves, 73space curves, 341arccosech, arccosh, arccoth, arcsech, arcsinh,arctanh, see hyperbolic functions, inversesArchimedean upthrust, 396, 410area element inCartesian coordinates, 188plane polars, 202area ofcircle, 71ellipse, 71, 207parallelogram, 223region, using multiple integrals, 191–193surfaces, 346as vector, 393–395, 408area, maximal enclosure, 779arg, argument of a complex number, 87Argand diagram, 84, 825argument, principle of the, 880arithmetic series, 117arithmetico-geometric series, 118arrays, see matricesassociated Laguerre equation, 535, 621–624as example of Sturm–Liouville equation, 566,622natural interval, 567, 622associated Laguerre polynomials L m n (x), 621as special case of confluent hypergeometricfunction, 634generating function, 623orthogonality, 622recurrence relations, 624Rodrigues’ formula, 622associated Legendre equation, 535, 587–593, 733,768general solution, 588as example of Sturm–Liouville equation, 566,590, 591general solution, 588natural interval, 567, 590, 591associated Legendre functions, 587–593of first kind Pl m (x), 588, 733, 768generating function, 592normalisation, 590orthogonality, 590, 591recurrence relations, 592Rodrigues’ formula, 588of second kind Q m l (x), 588associative law foradditionin a vector space of finite dimensionality,242in a vector space of infinite dimensionality,556of complex numbers, 86of matrices, 251of vectors, 213convolution, 447, 458group operations, 1043linear operators, 249multiplicationof a matrix by a scalar, 251ofavectorbyascalar,214of complex numbers, 88of matrices, 253multiplication by a scalarin a vector space of finite dimensionality,242in a vector space of infinite dimensionality,556atomic orbitals, 1115d-states, 1106, 1108, 1114p-states, 1106s-states, 1144auto-correlation functions, 450automorphism, 1061auxiliary equation, 493repeated roots, 493average value, see mean valueaxial vectors, 949backward differences, 1019basis functionsfor linear least squares estimation, 1273in a vector space of infinite dimensionality,556of a representation, 1078change in, 1084, 1087, 1092basis vectors, 217, 243, 929, 1078derivatives, 965–968Christoffel symbol Γ k ij , 965for particular irrep, 1106–1108, 1116linear dependence and independence, 217non-orthogonal, 245orthonormal, 244required properties, 217Bayes’ theorem, 1132Bernoulli equation, 477Bessel correction to variance estimate, 1248Bessel equation, 535, 602–607, 614, 615as example of Sturm–Liouville equation, 566natural interval, 608Bessel functions J ν (x), 602–614, 729, 738as special case of confluent hypergeometricfunction, 634generating function, 613graph of, 606integral relationships, 610integral representation, 613–614orthogonality, 608–6111306