Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXcentral limit theorem, 1194–1196central moments, see moments, centralcentre of a group, 1069centre of mass, 195of hemisphere, 195of semicircular lamina, 197centroid, 195of plane area, 195of plane curve, 197of triangle, 216CF, see complementary functionchain rule for functions ofone real variable, 46several real variables, 157change of basis, see similarity transformationschange of variablesand coordinate systems, 158–160in multiple integrals, 199–207evaluation of Gaussian integral, 202–204general properties, 206in RVD, 1150–1157character tables, 10933m or 32 or C 3v or S 3 , 1093, 1097, 1108, 1110,11174mm or C 4v or D 4 , 1102, 1106, 1108, 1113A 4 , 1116D 5 , 1116S 4 or 432 or O, 1114, 1115¯43m or T d , 1115construction of, 1100–1102quaternion, 1113characteristic equation, 280normal mode form, 319of recurrence relation, 499characteristic functions, see moment generatingfunctions (MGFs)characteristicsand boundary curves, 700multiple intersections, 700, 705and the existence of solutions, 699–705first-order equations, 699second-order equations, 703and equation type, 703characters, 1092–1096, 1100–1102and conjugacy classes, 1092, 1095counting irreps, 1095definition, 1092of product representation, 1104orthogonality properties, 1094, 1102summation rules, 1097charge (point), Dirac δ-function respresentation,441charged particle in electromagnetic fields, 370Chebyshev equation, 535, 595–602as example of Sturm–Liouville equation, 566,599general solution, 597natural interval, 567, 599polynomial solutions, 552Chebyshev functions, 595–602Chebyshev polynomialsof first kind T n (x), 596as special case of hypergeometric function,631generating function, 601graph of, 597normalisation, 600orthogonality, 599recurrence relations, 601Rodrigues’ formula, 599of second kind U n (x), 597generating function, 601graph of, 598normalisation, 600orthogonality, 599Rodrigues’ formula, 599chi-squared (χ 2 ) distribution, 1192and goodness of fit, 1297and likelihood-ratio test, 1283, 1292and multiple estimators, 1243percentage points tabulation, 1244test for correlation, 1301Cholesky separation, 313Christoffel symbol Γ k ij , 965–968from metric tensor, 966, 973circlearea of, 71equation for, 16circle of convergence, 831Clairaut equation, 483classes and equivalence relations, 1064closure of a group, 1043closure property of eigenfunctions of anHermitian operator, 563cofactor of a matrix element, 259column matrix, 250column vector, 250combinations (probability), 1133–1139common ratio in geometric series, 117commutation law for group elements, 1044commutative law foradditionin a vector space of finite dimensionality,242in a vector space of infinite dimensionality,556of complex numbers, 86of matrices, 251of vectors, 213complex scalar or dot product, 222convolution, 447, 458inner product, 244multiplicationofavectorbyascalar,214of complex numbers, 88scalar or dot product, 220commutatorof two matrices, 309of two operators, 653, 656comparison test, 1251308