Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXgeneral solution, 630natural interval, 567hypergeometric functions, 628–632contiguous relations, 632integral representation, 631recurrence relations, 632special cases, 630hypothesis testing, 1277–1298errors, first and second kind, 1280generalised likelihood ratio, 1282generalised likelihood-ratio test, 1281goodness of fit, 1296Neyman–Pearson test, 1280null, 1278power, 1280rejection region, 1279simple or composite, 1278statistical tests, 1278test statistic, 1278i, j, k (unit vectors), 219i, squarerootof−1, 84identity element of a group, 1043–1046uniqueness, 1043, 1045identity matrices, 254, 255identity operator, 249images, method of, see method of imagesimaginary part or term of a complex number, 83importance sampling, in Monte Carlo methods,1012improperintegrals, 70rotations, 946–948impulses, δ-function respresentation, 441incomplete gamma function, 639independent random variables, 1156, 1200index of a subgroup, 1066indices, of regular singular points, 540indicial equation, 540distinct roots with non-integral difference,540–542repeated roots, 542, 546, 547roots differ by integer, 542, 546induction, proof by, 31inequalitiesamongst integrals, 72Bessel, 246, 559Schwarz, 246, 559triangle, 246, 559inertia, see also moments of inertiamoments and products, 951tensor, 951inexact differentials, 155inexact equation, 473infinite integrals, 70contour integration, 862–867infinite series, see seriesinflectiongeneral points of, 52stationary points of, 50–52inhomogeneousboundary conditions, see boundaryconditions, homogeneous andinhomogeneousdifferential equations, 490simultaneous linear equations, 293inner product in a vector space, see also scalarproductof finite dimensionality, 244and Hermitian conjugate, 258commutativity, 244distributivity over addition, 244of infinite dimensionality, 557integral equationseigenfunctions, 817eigenvalues, 808, 816Fredholm, 805from differential equations, 803homogeneous, 805linear, 804first kind, 805second kind, 805nomenclature, 804singular, 805Volterra, 805integral equations, methods fordifferentiation, 812Fredholm theory, 815integral transforms, 809–812Fourier, 809–812Laplace, 810Neumann series, 813–815Schmidt–Hilbert theory, 816–819separable (degenerate) kernels, 807integral functions, 832nintegral test for convergence of series, 128integral transforms, see also Fourier transformsand Laplace transformsgeneral form, 459Hankel transforms, 459Mellin transforms, 459integrals, see also integrationcomplex, see complex integralsdefinite, 59double, see multiple integralsFourier transform of, 444improper, 70indefinite, 62inequalities, 72, 559infinite, 70Laplace transform of, 456limitscontaining variables, 188fixed, 59variable, 61line, see line integralsmultiple, see multiple integralsnon-zero, 1104of vectors, 339properties, 601318