Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXenergy spectrum and Fourier transforms, 450,451entire functions, 832nenvelopes, 173–175equations of, 174to a family of curves, 173epimorphism, 1061equilateral triangle, symmetries of, 1047, 1052,1081, 1110equivalence relations, 1064–1066, 1068and classes, 1064congruence, 1065–1067examples, 1070equivalence transformations, see similaritytransformationsequivalent representations, 1084–1086, 1099error function, erf(x), 640, 697, 748as special case of confluent hypergeometricfunction, 634error termsin Fourier series, 430in Taylor series, 139errors, first and second kind, 1280essential singularity, 838, 856estimation of eigenvalueslinear differential operator, 792–795Rayleigh–Ritz method, 327–329estimators (statistics), 1229best unbiased, 1232bias, 1231central confidence interval, 1237confidence interval, 1236confidence limits, 1236confidence region, 1241consistency, 1230efficiency, 1231maximum-likelihood, 1256minimum-variance, 1232standard error, 1234Euler equationdifferential, 504, 522trigonometric, 93Euler method, numerical, 1021Euler–Lagrange equation, 776special cases, 777–781even functions, see symmetric functionsevents, 1120complement of, 1121empty ∅, 1121intersection of ∩, 1120mutually exclusive, 1129statistically independent, 1129union of ∪, 1121exact differentials, 155exact equations, 472, 505condition for, 472non-linear, 519expectation values, see probability distributions,meanexponential distribution, 1190from Poisson, 1190MGF, 1191exponential functionMaclaurin series for, 140of a complex variable, 92, 833relation with hyperbolic functions, 102F-distribution (Fisher), 1290–1296critical points table, 1295logarithmic form, 1296Fabry–Pérot interferometer, 146factorial function, general, 636factorisation, of a polynomial equation, 7faithful representation, 1083, 1098Fermat’s principle, 787, 798Fermi–Dirac statistics, 1138Fibonacci series, 525field lines and complex potentials, 872fieldsconservative, 387–389scalar, 347tensor, 954vector, 347fields, electrostatic, see electrostatic fields andpotentialsfields, gravitational, see gravitational fields andpotentialsfinite differences, 1019central, 1019for differential equations, 1020–1023forward and backward, 1019from Taylor series, 1019, 1026schemes for differential equations, 1030–1032finite groups, 1043first law of thermodynamics, 176first-order differential equations, see ordinarydifferential equationsFisher distribution, see F-distribution (Fisher)Fisher matrix, 1241, 1268Fisher’s inequality, 1232, 1233fluidsArchimedean upthrust, 396, 410complex velocity potential, 873continuity equation, 404cylinder in uniform flow, 874flow, 873flux, 395, 875irrotational flow, 353sources and sinks, 404, 873stagnation points, 873velocity potential, 409, 679vortex flow, 408, 874forward differences, 1019Fourier cosine transforms, 446Fourier series, 415–432and separation of variables, 719–722, 724coefficients, 417–419, 425complex, 424differentiation, 424Dirichlet conditions, 4151314