Handbook of mathematics for engineers and scienteists part 216 potx

INDEX 1473 function (continued) probability density, 1040 probability-generating, of random variable, 1047 psi, 944 random variables, 1041 range, 236 rational, 276 real variable, 236 regression, 1063, 1066 regular at point, 403 Riemann, 628, 629, 783 sample, 1072 sign, xxxi sine, 24, 25 sine, hyperbolic, xxxi, 35 single-valued, 236, 402 spectral of stationary process, 1075 square integrable, 314 stationary value, 220 stream, 1295 strictly concave, 245 strictly convex, 245 Struve, 448, 449 symmetric elementary, 163 tangent, xxxi, 24, 26, 56, 368, 380 tangent, hyperbolic, xxxi, 35 theoretical distribution, 1082 theta, 978 total variation, 247 Weber parabolic cylinder, 967 Weierstrass, 998, 1000 Weierstrass elliptic, 976 Whittaker, 960 zero, 413 Zhukovskii, 404, 424 functional, 991 bilinear, 992 Bolza, 1001 classical integral, 993 differentiable at point, 992 first variation, 992 increment, 992 linear, 213 quadratic, 992 quadratic definite positive, 993 second variation, 992, 993 strong minimum, 991 strongly positive, 993 variation, 992 weak minimum, 991 functional differential equations, 682, 684, 688, 931 functional equation Abel, 914 Babbage, 917, 1416, 1423 bilinear, 930, 1446 Cantor first, 1442 Cantor second, 1449 Cauchy, 1438 Cauchy exponential, 1443 functional equation (continued) Cauchy logarithmic, 1439 Cauchy power, 1444 cyclic, 920 D’Alembert, 1443 Darboux, 458 difference, 873 difference, logistic, 875 Gauss, 1445 Gauss, generalized, 1445 homogeneous function, 1441 information theory, basic, 1440 involutory functions, 917, 1423 Jensen, 1438 Jensen, generalized, 1449 linear homogeneous, 910 linear nonhomogeneous, 912 Lobachevsky, 1444 logarithmic, Cauchy, 1439 Pexider, 925, 1440 reciprocal, 919 Riccati, difference, 875, 918, 1428, 1429 Schr ¨ oder–Koenigs, 911 self-similar solutions, 1442 traveling-wave, 1441 functional equations, 705–707, 1409, 1414–1421, 1446 containing iterations of unknown function, 917 in several independent variables, 922, 1438 linear, 1414–1421 functional mean, 13 functional separable solution, 697 special, 697 functional separation of variables, 697 functional series, 348, 1120 criteria of uniform convergence, 349 Dirichlet criterion, 350 pointwise convergence, 348 properties, 350 remainder, 349 uniform convergence, 349 functional sums, 1116 functions coordinate, 847, 851 dependent, 267 elementary, 19–41 independent, 267 of same order, 243 small infinitely equivalent, 243 special, 939–990 fundamental solution, 599, 615 of Cauchy problem, 616, 617 fundamental solutions for system of ODEs, 539 fundamental system of solutions, 477, 478, 522, 523 of homogeneous system, 198 fundamental theorem of algebra, 162 G Gakhov first theorem, 432 Gakhov second theorem, 433 1474 INDEX Galerkin method, 508 Galileo transformation, 718 game, 1024 antagonistic, 1024 coalition, 1024 finite, 1024 infinite, 1024 lower price, 1025 theory, 1024 two-person, 1024 upper price, 1025 with saddle point, 1025 game price in mixed strategies, 1026 game theory, 1024 game value, 1025 “games with nature”, 1024 gamma function, xxxii, 943, 1213 incomplete, xxxii, 946 logarithmic derivative, 949, 954 gas dynamics, model equation, 1254 Gauss derivation formulas, 396 Gauss equation, 1445 Gauss error function, xxxi Gauss formula, 396, 808 Gauss theorem, 335 Gauss transform, 449 Gauss transformation, 974 Gauss–Jordan elimination, 201 Gaussian coordinates, 387 Gaussian curvature, 394 Gaussian elimination, 199 Gaussian elimination method, backward substitution, 200 Gaussian hypergeometric equation, 960, 1216 Gegenbauer functions, 1218 Gegenbauer polynomials, 987 Gel’fand–Levitan–Marchenko integral equation, 1328, 1333 general equation of plane, 124 of quadric, 148 of second-order curve, 107 of second-order hypersurface, 220 of straight line, 90 of straight line in space, 132 of straight line on plane, 90 general form of Laplace integral, 292 of Schr ¨ odinger equation, 1311 of second-order finite-difference equation, 1436 of second-order nonlinear difference equation, 1436 general fourth-degree equation, 160 general integral, 570, 571 general linear group, 228 general mathematical programming problem, 1012 general orthogonal curvilinear coordinates, 1196 general Riccati equation, 461, 462, 1209, 1210 general solution of difference equation, 874, 884 general solution (continued) of first-order PDEs, 553 of linear algebraic system, 203 of linear hyperbolic equations, 600–602 of linear ODEs, 472, 514, 518 of ODEs, 453, 488 general square system of linear algebraic equations, 199 general system of m linear algebraic equations with n unknown quantities, 203 generalized Abel equation, 806 of second kind, 1393 generalized Burgers–Korteweg–de Vries equation, 709 generalized Cauchy problem, 627 generalized Chebyshev inequalities, 15 generalized classical method of characteristics, 582 generalized Dirichlet problem, 428 generalized Ermakov criterion of convergence of series, 340 generalized Ermakov system, 1242 generalized Gauss equation, 1445 generalized homogeneous equation, 490, 526, 1210, 1224 generalized Jensen equation, 1449 generalized Jensen inequalities, 246 generalized Laguerre polynomials, 983 generalized permutation matrix, 168 generalized reciprocal equation, 160 generalized self-similar solution, 677 generalized separable solution, 681, 682 generalized separation of variables, 681 generalized similarity reduction, 746 generalized solution, 563 Hopf formula, 565 of first-order PDE, 563, 568 regular point, 580 singular point, 580 stable, 569 generalized telegraph equation, 626 generalized traveling-wave solution, 697 generalized triangle inequality, 343 generalized variance of n-dimensional distribution, 1065 generalized viscosity solutions, 579, 580 generating function, 815, 1047 of numerical sequence, 1047 generator, 65, 67 group, 717, 730 infinitesimal, 717 generic term of sequence, 237 of series, 337 geodesic arc, 70 geodesic curvature, 392 geodesic curvature vector, 392 geometric definition of nondegenerate second- order curve, 111 geometric distribution, 1048 geometric mean, 13 of functions, 297 INDEX 1475 geometric multiplicity of eigenvalue, 209 geometric probability, 1034 geometric programming, 1027 geometric progression, 11 geometric sequence, 11 geometric series, 11 geometry analytic, 77 differential, 367 intrinsic, of surface, 391 spherical, 70 Gergonne point, 47 Goodness-of-fit tests, 1098, 1101 gorge ellipse, 145 Goursat problem, 592, 629 Gram determinant, 193 Gram–Schmidt orthogonalization, 194 graph of function, 84, 236 gravitational field potential, 328 great circle, 68, 70 spherical geometry, 70 greatest common divisor, 4 Green first identities, 335 Green formula, 428 Green function, 428, 623, 631, 633–637 admitting incomplete separation of variables, 640 for elliptic equations, 631–634, 637 for hyperbolic equations, 623–625, 636 for linear ordinary differential equation, 481 for parabolic equations, 618–622, 635 in terms of infinite series, 639 Green second identities, 335 Green theorem, 335 group abelian, 226 automorphism, 227 commutative, 226 completely reducible representation, 231 conjugate element, 231 continuous point, 730 equivalent representations, 230 faithful representation, 230 invariant, 731 Lie, 730 linear representation, 230 of derivation formulas, second, 396 of events, complete, 1031 of incompatible events, complete, 1032 of leading terms of equation, 220 of linear transformations, 228 of mirror symmetry, 229 of modulo 2 residues, 226 of orthogonal transformations, 229 orthogonal, 229 quotient, 227 special linear, 228 special orthogonal, 229 trivial representation, 230 group analysis of differential equations, 716 group element, character, 231 group generator, 717, 730, 732 group invariant, 731 group theory, 225 groups basic properties, 225 cyclic, 229 isomorphic, 227 permutation, 226 unitary, 229 growth exponent, 436 H half-plane left, 79 lower, 79 right, 79 upper, 79 half-side theorem, 73 Hamilton operator, 272 Hamilton–Jacobi equation, 576 Hamiltonian, 576 Hankel functions, 952 Hankel transform, 446, 449, 1400 Hardy transform, 449 harmonic function, 404 harmonic mean, 13 of functions, 297 harmonic series, 339 Hartley transform, 449 heat equation linear, 585, 1267 linear, nonhomogeneous, 1268 nonlinear, 1318 stationary, anisotropic, 1323 stationary, with nonlinear source, 1320 with axial symmetry, 1270 with central symmetry, 1272 Heaviside unit step function, 618 Helmholtz equation, 1289 Hermite functions, 986 Hermite polynomials, xxxi, 957, 958, 985 Hermitian matrix, 168 Hermitian operator, 206 negative, 207 nonnegative, 207 nonpositive, 207 positive, 207 Heron formula, 48 higher-order derivatives of elementary functions, 255 higher-order linear difference equations, 1425 Hilbert boundary value problem, 432 Hilbert example, 997 Hilbert space, 196 Hilbert transform, 449, 1396, 1400 Hilbert–Privalov boundary value problem, 432 Hilbert–Schmidt theorem, 837 histogram, 1086 frequency, 1086 relative frequency, 1086 1476 INDEX hodograph transformation for gas dynamic system, 784 for ODEs, 455 for PDEs, 657, 658 H ¨ older condition, 310, 430 H ¨ older constant, 310 H ¨ older exponent, 310 H ¨ older inequality, 14, 297, 343 H ¨ older–Jensen inequality, 15 homogeneity, order, 1441 homogeneous boundary condition, 591 homogeneous equations linear difference, 881, 885, 889, 1421, 1425 linear difference, constant-coefficient, 1409, 1422, 1426 linear integral, 810, 829, 834 ordinary differential, 456, 490, 526, 1207, 1224 ordinary differential, generalized, 457, 490, 526, 1224 ordinary differential, linear, 472, 514 partial differential, elliptic, 592 partial differential, hyperbolic, 592 partial differential, parabolic, 591 homogeneous function, 1441 homogeneous initial condition, 591 homogeneous linear algebraic system, 203 general solution, 203 homogeneous linear integral equation, 810, 829 homogeneous linear ODEs, 472, 514 of higher-order ODEs, 532 homogeneous Monge–Amp ` ere equation, 668, 1326 homogeneous system of linear algebraic equations, 197 of linear ODEs, 539 of linear ODEs, superposition principle, 528, 539 homomorphic image, 226 homomorphisms, 226 “hoof”, 67 Hopf formula for generalized solution, 565 Hopf–Cole transformation, 666, 1308 Horner scheme, 156 Hurwitz functions, 893 hyperbola, 101 arm, 101 asymptotes, 101 canonical equation, 101 center, 102 conjugate diameters, 104 diameter, 104 directrix, 102 eccentricity, 102 equilateral, 101 focal axis, 101 focal distance, 102 focal parameter, 102 focal properties, 102 focus, 102 focus-directrix property, 103 imaginary axis, 101 hyperbola (continued) imaginary semiaxis, 102 in polar coordinate system, 104 left arm, 101 left directrix, 102 left focus, 102 linear eccentricity, 102 numerical eccentricity, 102 optical property, 103 parameter, 102 real semiaxes, 102 right arm, 101 right directrix, 102 right focus, 102 vertex, 102 hyperbolic arccosine, xxx hyperbolic arccotangent, xxx hyperbolic arcsine, xxx hyperbolic arctangent, xxx hyperbolic cosine, xxx, 35 hyperbolic cotangent, xxx, 35 hyperbolic equation boundary value problems, 623 canonical form, 587 first canonical form, 587 second canonical form, 587 hyperbolic equations, 585, 590, 647, 648, 742 Cauchy problem, 617 linear, 587, 617, 623–630, 636, 647, 648, 1278–1284 nonlinear, 653, 654, 1312–1317 hyperbolic functions, 34 addition formulas, 36 differentiation formulas, 38 expansion in power series, 39 integration formulas, 38 inverse, 39 of half argument, 38 of multiple argument, 38 properties, 36 hyperbolic paraboloid, 146 hyperbolic point, 394, 536 hyperbolic sine, xxxi, 35 hyperbolic spiral, 86, 368 hyperbolic system of PDEs, 781, 790 hyperbolic tangent, xxxi, 35 hyperbolic type of second-order curves, 108 hyperboloid, 144, 224 of revolution, 145 one-sheeted, 144 real semiaxes, 144 two-sheeted, 144 hypergeometric distribution, 1049 hypergeometric equation, 960, 1218 degenerate, 1215 hypergeometric functions, 960 hypergeometric series, 960, 1218 hypersurface, 220 center, 222 central, 222 noncentral, 224 semiaxis, 223 INDEX 1477 hypotenuse, 50, 72 hypothesis, 1094 alternative, 1094 composite, 1094 nonparametric, 1094 null, 1094 parametric, 1094 simple, 1094 statistical, 1094 I identity Abel, 902 Parseval, 361 identity matrix, 168 identity operator, 204 image, homomorphic, 226 imaginary axis, 401 of hyperbola, 101 imaginary ellipsoid, 224 imaginary part, 399 imaginary semiaxis of hyperbola, 102 of one-sheeted hyperboloid, 144 of two-sheeted hyperboloid, 144 imaginary unit, 399 impossible event, 1031 improper integral, 301 improper integrals involving logarithmic functions, 305 involving power functions, 304 involving trigonometric functions, 305 of unbounded functions, 309 sufficient conditions for convergence, 302 improper orthogonal transformations, 229 improper subspace, 190 incenter, 47 incenter point, 46 incircle, 46 of triangle, 46, 47 incompatible events, 1031 incomplete beta function, 947 incomplete cubic equation, 158 incomplete elliptic integral, 970 incomplete equation of plane, 124 incomplete gamma function, xxxii, 946, 957, 958 incomplete separation of variables, 597, 616, 640 incomplete system of eigenfunctions, 836 inconsistent system of linear algebraic equations, 197 increasing function, 236, 238 increment of argument, 992 of functional, 992 partial, 264 indefinite integrals, 273, 1129 involving exponential functions, 1137 involving hyperbolic functions, 1137 involving inverse trigonometric functions, 1147 involving irrational functions, 1134 involving logarithmic functions, 1140 indefinite integrals (continued) involving rational functions, 1129 involving trigonometric functions, 1142 properties, 274 independent random variables, 1060, 1067 independent trials, 1037 independent variable, 236 index characteristic, 1221 Fuchs, 749 of inertia, 218 of inertia, negative, 218 of inertia, positive, 218 of matrix, 182 of multiplication, 12 of summation, 12 inequality Bunyakovsky, 296 Cauchy, 14, 296, 343, 410 Cauchy–Bunyakovski, 14, 296, 343 Cauchy–Schwarz, 121, 193, 1043 Cauchy–Schwarz–Bunyakovski, 14, 314 Chebyshev, 14, 297, 1070 Chebyshev, generalized, 15 Cram ´ er–Rao, 1089 H ¨ older, 14, 297, 343 H ¨ older–Jensen, 15 Jensen, 245, 297 Markov, 1070 Minkowski, 14, 121, 193, 296, 343 Steklov, 297 Sylvester, 178 triangle, 44, 73, 193, 314 triangle, generalized, 14, 343 infimum, xxxi infinite game, 1024 infinite matrix, 167 infinite product, 346, 348 absolute convergence, 346 convergent, 346 divergent, 346 infinite products convergence criteria, 347 with positive factors, 347 infinite series, 1116 infinite-dimensional linear space, 189 infinitely large function, 243 infinitely large quantity, 239 infinitely large sequence, 239 infinitely small function, 242, 243 infinitely small quantity, 239 infi nitesimal generator, 717 infinitesimal operator, 716, 717, 730 inflection point, 258 information, Fisher, 1089 information theory, basic equation, 1438 initial condition for first-order ODEs, 454 for parabolic equations, 591 homogeneous, 591 1478 INDEX initial conditions for hyperbolic equations, 592 for ODEs, 488, 524 for Riemann problem, 785, 793 homogeneous, 592 initial moment of multivariate random variable, 1065 of two-dimensional random variable, 1060 initial value problem, 580, 785 injective linear operator, 205 inner product, 192, 314, 840 inscribed angle, 56 inscribed polygon, in circle, 51 integer number, 3 integer power of complex number, 399 of linear operator, 205 of matrix, 172 integrable combinations, 545 integrable function, 286, 312, 314 integral absolutely convergent, 303 Cauchy-type (singular), 310, 409, 430 closed path, 331 complete, 570 curvilinear, 329 definite, 286 Dini, 1287 Dirichlet–Mehler, 963 divergent, 301 Duhamel, 516 elliptic, complete, of first kind, 969 elliptic, complete, of second kind, 969 elliptic, of first kind, 970 elliptic, of second kind, 970 elliptic, of third kind, 971 exponential, 958 fractional, 449 Fresnel, 304 Frullani, 307 general, 571 improper, 301 indefinite, 273 involving power-law functions, 1147 Laplace, 418, 521, 963 Laplace, of general form, 292 Laplace, power, 294 Laplace, power, multiple, 328 Laplace, special form, 291 line, of first kind, 329 line, of second kind, 330, 331 logarithmic, 958 multiple Laplace-type, 328 of differential binomial, 281 of function of complex variable, 407 path, 329 Poisson, 428, 1287 Poisson, for half-plane, 429 potential-type, 291 iterated, 319 Riemann, 286 integral (continued) Schwarz, 428 Schwarz, for half-plane, 429 singular (Cauchy-type), 310, 311, 430, 571 Stieltjes, 312 surface, of first kind, 332 surface, of second kind, 333 with infinite limits, 301 integral equation Abel, 1385 Abel, generalized, 1386 Abel, of second kind, 1392 Abel, of second kind, generalized, 1393 existence and uniqueness of solution, 802 first kind, with constant integration limit, 824 first kind, with variable integration limit, 801 Fredholm, of first kind, 824, 825 Fredholm, of second kind, 829 Gel’fand–Levitan–Marchenko, 1328, 1333 homogeneous, 810, 829 kernel, 801, 825, 829 Lalesco–Picard, 1403 linear, 801, 810, 824, 829 nonhomogeneous, 810, 829 nonlinear, 856 nonlinear, with constant integration limits, 856, 863 nonlinear, with degenerate kernel, 863 nonlinear, with variable integration limit, 856 parameter, 829 regularization methods, 827 right-hand side, 810 Schl ¨ omilch, 1400 second kind, with constant integration limits, 829 second kind, with variable integration limit, 810 skew-symmetric, 840 symmetric, 835 Tricomi, 1403 Urysohn, nonlinear, 856 Urysohn, with degenerate kernel, 863, 864 Volterra, nonlinear, 856, 858–861 Volterra, of first kind, 801, 807 Volterra, of second kind, 807, 810 Volterra, with quadratic nonlinearity, 856 Wiener–Hopf, of first kind, 825, 1401 Wiener–Hopf, of second kind, 829, 1406 with degenerate kernel, 802, 803, 811, 812 with difference kernel, 804, 813, 825, 829, 841 with quadratic nonlinearity, 865 with weak singularity of second kind, 829 integral equations, 801–871, 1385–1406 first kind, with constant limits of integration, 824, 1396–1400 first kind, with variable limit of integration, 801, 1385–1390 first kind, with weak singularity, 825 linear, 801–855, 1385–1406 method of differentiating, 866 second kind, with constant limits of integra- tion, 829, 1401–1408 INDEX 1479 integral equations (continued) second kind, with variable limit of integration, 810, 1391–1395 integral formulas of vector calculus, 334 integral sum, 286, 317 Stieltjes, 312 integral theorem of de Moivre–Laplace, 1038 integral transform Bochner, 449 Euler, first kind, 449 Euler, second kind, 450 Fourier, 443, 449 Fourier, asymmetric form, 444 Fourier, inverse, 443 Fourier, n-dimensional, 445 Fourier, standard form, 443 Fourier cosine, 445, 449 Fourier sine, 446, 449 Gauss, 449 Hankel, 446, 449 Hardy, 449 Hartley, 449 Hilbert, 449 kernel, 435 Kontorovich–Lebedev, 448, 449 Laplace, 436, 449 Laplace, inverse, 437 Laplace, two-sided, 449 Meijer, 447, 449 Meler–Fock, 449 Mellin, 441, 449 Mellin, inverse, 442 Weber, 449 Weierstrass, 449 integral transforms, 435, 1155 general form, 435 inversion formulas, 435 method for linear PDEs, 611 integrals, tables, 1129 integrals of motion, 766 integrand (function), 273 integrating factor, 459 of Pfaffian equation, 773 integration by parts, 275 by parts, repeated, 275 of asymptotic series, 365 of exponential functions, 281 of hyperbolic functions, 281 of improper fractions, 278 of irrational functions, 279 of proper fraction, 277 of rational functions, 276 of trigonometric functions, 282 intercept equation of plane, 125 intercept-intercept equation of straight line, 91 intergroup sample variance, 1108 interior angle, 43 of polygon, 51 interpolation nodes, 808 interpolation polynomial, 186 intersection of events, 1031 of four planes, 139 of line and plane, 140 of subspaces, 190 of three planes, 139 interval, 235 closed, xxx, 235 confidence, 1093 estimator, 1088, 1093 half-open, xxx, 234 of convergence of power series, 351 open, xxx, 235 series, 1083 intrinsic equations of curve, 388 intrinsic geometry of surface, 391, 395 invariance condition, 719, 720, 732 invariant, 222, 731 of group, 731 of infinitesimal operator, 717 of linear operator, 211 of operator, 717, 731 of quadric, 148 of second-order curve, 107 universal, 731 invariant condition, 731 invariant representation, 230 invariant solution, 669, 674, 716, 724 invariant subspace, 205, 230 invariant surface condition, 732 inverse element, 225 inverse Fourier transform, 443, 614 inverse function, 237 inverse hyperbolic cosine, 40 inverse hyperbolic cotangent, 40 inverse hyperbolic functions, 39 differentiation formulas, 41 expansion in power series, 42 integration formulas, 41 properties, 41 inverse hyperbolic sine, 39 inverse hyperbolic tangent, 40 inverse Laplace transform, 437, 611 tables, 1162 inverse matrix, 171 left, 171 right, 171 inverse Mellin transform, 442 tables, 1190 inverse of mapping, 226 inverse operator, 205 inverse scattering problem, 760 method, 755 inverse transform, 435 Fourier, 443 Fourier cosine, 445 Fourier sine, 446 Hankel, 446 Kontorovich–Lebedev, 448 Laplace, 437 Meijer, 447 . 230 of derivation formulas, second, 396 of events, complete, 1031 of incompatible events, complete, 1032 of leading terms of equation, 220 of linear transformations, 228 of mirror symmetry, 229 of. 773 integration by parts, 275 by parts, repeated, 275 of asymptotic series, 365 of exponential functions, 281 of hyperbolic functions, 281 of improper fractions, 278 of irrational functions, 279 of proper. transforms, 435, 1155 general form, 435 inversion formulas, 435 method for linear PDEs, 611 integrals, tables, 1129 integrals of motion, 766 integrand (function), 273 integrating factor, 459 of