Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXcomplement, 1121probability for, 1125complementary equation, 490complementary error function, 640complementary function (CF), 491for ODE, 492partially known, 506repeated roots of auxiliary equation, 493completeness ofbasis vectors, 243eigenfunctions of an Hermitian operator, 560,563eigenvectors of a normal matrix, 275spherical harmonics Yl m (θ, φ), 594completing the squareas a means of integration, 66for quadratic equations, 35for quadratic forms, 1206to evaluate Gaussian integral, 436, 749complex conjugatez ∗ , of complex number, 89–91, 829of a matrix, 256–258of scalar or dot product, 222properties of, 90complex exponential function, 92, 833complex Fourier series, 424complex integrals, 845–849, see also zeros of afunction of a complex variable and contourintegrationAiry integrals, 890–894Cauchy integrals, 851–853Cauchy’s theorem, 849definition, 845Jordan’s lemma, 864Morera’s theorem, 851of z −1 , 846principal value, 864residue theorem, 858–860WKB methods, 895–905complex logarithms, 99, 834principal value of, 100, 834complex numbers, 83–114addition and subtraction of, 85applications to differentiation and integration,101argument of, 87associativity ofaddition, 86multiplication, 88commutativity ofaddition, 86multiplication, 88complex conjugate of, see complex conjugatecomponents of, 84de Moivre’s theorem, see de Moivre’s theoremdivision of, 91, 94from roots of polynomial equations, 83imaginary part of, 83modulus of, 87multiplication of, 88, 94as rotation in the Argand diagram, 88notation, 84polar representation of, 92–95real part of, 83trigonometric representation of, 93complex potentials, 871–876and fluid flow, 873equipotentials and field lines, 872for circular and elliptic cylinders, 876for parallel cylinders, 921for plates, 877–879, 921for strip, 921for wedges, 878under conformal transformations, 876–879complex power series, 133complex powers, 99complex variables, see functions of a complexvariable and power series in a complexvariable and complex integralscomponentsof a complex number, 84of a vector, 217in a non-orthogonal basis, 234uniqueness, 243conditional (constrained) variation, 785–787conditional convergence, 124conditional distributions, 1198conditional probability, see probability,conditionalconesurface area of, 74volume of, 75confidence interval, 1236confidence region, 1241confluence process, 634confluent hypergeometric equation, 535, 633as example of Sturm–Liouville equation, 566general solution, 633confluent hypergeometric functions, 633contiguous relations, 635integral representation, 634recurrence relations, 635special cases, 634conformal transformations (mappings), 839–879applications, 876–879examples, 842–844properties, 839–842Schwarz–Christoffel transformation, 843congruence, 1065conic sections, 15eccentricity, 17parametric forms, 17standard forms, 16conjugacy classes, 1068–1070element in a class by itself, 1068conjugate roots of polynomial equations, 99connectivity of regions, 383conservative fields, 387–389necessary and sufficient conditions, 387–389potential (function), 3891309