Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXconsistency, of estimator, 1230constant coefficients in ODE, 492–503auxiliary equation, 493constants of integration, 62, 468constrained variation, 785–787constraints, stationary values under, seeLagrange undetermined multiplerscontinuity correction for discrete RV, 1186continuity equation, 404contour integration, 861–867, 887infinite integrals, 862–867inverse Laplace transforms, 884–887residue theorem, 858–867sinusoidal functions, 861summing series, 882contraction of tensors, 939contradiction, proof by, 32–34contravariantbasis vectors, 961derivative, 965components of tensor, 956definition, 961control variates, in Monte Carlo methods, 1013convergence of infinite series, 831absolute, 124, 831complex power series, 133conditional, 124necessary condition, 125power series, 132under various manipulations, see powerseries, manipulationratio test, 832rearrangement of terms, 124tests for convergence, 125–131alternating series test, 130comparison test, 125grouping terms, 129integral test, 128quotient test, 127ratio comparison test, 127ratio test (D’Alembert), 126, 132root test (Cauchy), 129, 831convergence of numerical iteration schemes,992–994convolutionFourier tranforms, see Fourier transforms,convolutionLaplace tranforms, see Laplace transforms,convolutionconvolution theoremFourier transforms, 448Laplace transforms, 457coordinate geometry, 15–18conic sections, 15straight line, 15coordinate systems, see Cartesian, curvilinear,cylindrical polar, plane polar and sphericalpolar coordinatescoordinate transformationsand integrals, see change of variablesand matrices, see similarity transformationsgeneral, 960–965relative tensors, 963tensor transformations, 962weight, 964orthogonal, 932coplanar vectors, 225Cornu spiral, 914correlation functions, 449–451auto-correlation, 450cross-correlation, 449energy spectrum, 450Parseval’s theorem, 451Wiener–Kinchin theorem, 450correlation matrix, of sample, 1229correlation of bivariate distributions, 1200–1207correlation of sample data, 1229correlation, chi-squared test, 1301correspondence principle in quantum mechanics,1215cosets and congruence, 1065cosh, hyperbolic cosine, 102, 833, see alsohyperbolic functionscosine, cos(x)in terms of exponential functions, 102Maclaurin series for, 140orthogonality relations, 417counting irreps, see characters, counting irrepscoupled pendulums, 329, 331covariance matrixof linear least squares estimators, 1274of sample, 1229covariance of bivariate distributions, 1200–1207covariance of sample data, 1229covariantbasis vector, 961derivative, 965components of tensor, 956definition, 961derivative, 968of scalar, 971semi-colon notation, 969differentiation, 968–971CPF, see probability functions, cumulativeCramér–Rao (Fisher’s) inequality, 1232, 1233Cramer determinant, 299Cramer’s rule, 299cross product, see vector productcross-correlation functions, 449crystal lattice, 148crystal point groups, 1082cube roots of unity, 98cube, rotational symmetries of, 1114cumulants, 1166curl of a vector field, 353as a determinant, 353as integral, 398, 400curl curl, 356in curvilinear coordinates, 368in cylindrical polars, 3601310