Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXstandard deviation σ, 1146variance σ 2 , 1146probability functions (PFs), 1139cumulative (CPFs), 1139, 1141density functions (PDFs), 1140probability generating functions (PGFs),1157–1162and MGF, 1163binomial, 1161definition, 1158geometric, 1159mean and variance, 1158Poisson, 1158sums of RV, 1161trials, 1158variable sums of RV, 1161product rule for differentiation, 44–46, 48–50products of inertia, 951projection operators for irreps, 1107, 1116projection tensors, 979proper rotations, 946proper subgroups, 1061pseudoscalars, 947, 950pseudotensors, 946–950, 964pseudovectors, 946–950Q l (x), see Legendre polynomialsQ m l (x), see associated Legendre functionsquadratic equationscomplex roots of, 83properties of roots, 10roots of, 2quadratic forms, 288–292completing the square, 1206positive definite and semi-definite, 290quadratic surfaces, 292removing cross terms, 289stationary properties of eigenvectors, 290quantum mechanics, from classical mechanics,657quantum operators, see operators (quantum)quartiles, of RVD, 1145quaternion group, 1073, 1113quotient law for tensors, 939–941quotient rule for differentiation, 47quotient test for series, 127radius of convergence, 133, 831radius of curvatureof plane curves, 53of space curves, 342radius of torsion of space curves, 343random number generation, 1017random numbers, non-uniform distribution,1035random variable distributions, see probabilitydistributionsrandom variables (RV), 1119, 1139–1143continuous, 1140–1143dependent, 1196–1205discrete, 1139independent, 1156, 1200sums of, 1160–1162uncorrelated, 1200range of a matrix, 293rank of matrices, 267and determinants, 267and linear dependence, 267rank of tensors, see order of tensorrate of change of a function ofone real variable, 41several real variables, 153–155ratio comparison test, 127ratio test (D’Alembert), 126, 832in convergence of power series, 132ratio theorem, 215and centroid of a triangle, 216Rayleigh–Ritz method, 327–329, 800real part or term of a complex number, 83real roots, of a polynomial equation, 1rearrangement methods for algebraic equations,987reciprocal vectors, 233, 366, 955, 959reciprocity relation for partial derivatives, 157rectangular distribution, 1194Fourier transform of, 442recurrence relations (functions), 585, 611associated Laguerre polynomials, 624associated Legendre functions, 592Chebyshev polynomials, 601confluent hypergeometric functions, 635Hermite polynomials, 628hypergeometric functions, 632Laguerre polynomials, 620Legendre polynomials, 586recurrence relations (series), 496–501characteristic equation, 499coefficients, 536, 538, 999first-order, 497second-order, 499higher-order, 501reducible representations, 1084, 1086reduction formulae for integrals, 69reflectionsand improper rotations, 946as symmetry operations, 1041reflexivity, and equivalence relations, 1064regular functions, see analytic functionsregular representations, 1097, 1110regular singular points, 534, 538–540relative velocities, 218remainder term in Taylor series, 138repeated roots of auxiliary equation, 493representation, 1076definition, 1082dimension of, 1078, 1082equivalent, 1084–1086faithful, 1083, 1098generation of, 1078–1084, 1112irreducible, see irreps1327