INDEX 1487 multiplicity of eigenvalue algebraic, 184, 209 geometric, 209 multiplier, Lagrange, 270 multiply connected domain, 409 multivalued function, 236, 402 multivariate random variable initial moment, 1065 moment, 1065 N n-dimensional coordinate space, 188 n-dimensional distribution, generalized variance, 1065 n-dimensional ellipsoid, 224 n-dimensional Fourier transform, 445 n-dimensional linear space, 189 n-dimensional random variable, 1065 n-dimensional random vector, 1057 n-fold singular point, 371 n-person game, 1024 n-sided pyramid, 63 N-soliton solution, 765 Korteweg–de Vries equation, 764 Nagel point, 47 Napierian base, 23 Napierian logatithm, see natural logarithm natural logarithm, xxxi, 9, 22, 23, 238 base, 9, 23 natural number, 3 natural trihedral, 382 necessary condition for extremum, 269, 1002, 1006, 1009 for solvability of Neumann problem, 429 negative binomial distribution, 1050 negative definite quadratic form, 216 negative Hermitian operator, 207 negative index of inertia of quadratic form, 218 negative symmetric operator, 207 neighborhood first-order, 991 of point at infinity, 402 of point on complex plane, 402 strong, 991 weak, 991 zero-order, 991 Neper analogs, 73 Neper rules, 75 net, coordinate, 387 Neumann function, 947 Neumann problem, 429 for elliptic equations, 593, 1330 necessary condition for solvability, 429 Neumann series, 823, 833 neutral element, 225 Newell–Whitehead equation, 733 Newton method, 164 Newton–Kantorovich method, 859, 868 Newton–Leibniz formula, 288 for function of complex variable, 408 Newton–Raphson method, 262 Neyman–Fisher theorem, 1090 Neyman–Pearson lemma, 1097 nilpotent matrix, 172 nine-point circle, 49 node, 372 Chebyshev, 523 interpolation, 808 quadrature, 808 stable, 535, 538 unstable, 536 Noetherian symmetry, 767 of Lagrangian, 768 nonbasic variable, 1015 nonclassical method of symmetry reductions, 732, 747 nonclassical symmetries, 733 nondecreasing function, 236 nondegenerate bilinear form, 215 nondegenerate linear operator, 205 nondegenerate matrix, 171 nondegenerate quadratic form, 216 nonhomogeneous nth-order constant-coefficient linear difference equation, 1427 nonhomogeneous nth-order linear difference equation, 1426 nonhomogeneous biharmonic equation, 1298 nonhomogeneous first-order constant-coefficient linear difference equation, 1411 nonhomogeneous higher-order linear systems, D’Alembert method, 533 nonhomogeneous integral equation, 810, 829 nonhomogeneous linear boundary value problems, 620, 624 nonhomogeneous linear difference equations, 882 nonhomogeneous linear ODEs, 474, 515 nonhomogeneous linear PDEs, 598 nonhomogeneous Monge–Amp ` ere equation, 1326 nonhomogeneous second-order constant-coefficient linear difference equation, 1422 nonhomogeneous second-order linear difference equation, 1421 nonhomogeneous system of linear algebraic equations, 197 of linear first-order ODEs, 531, 540 superposition principle, 541 nonincreasing function, 236 nonlinear nth-order difference equation, 919 nonlinear difference equation, 884, 918, 1428 of general form, 1437 nonlinear diffusion equation with cubic source, 753 nonlinear elliptic equations, 1318 nonlinear first-order partial differential equation with two independent variables, 570 nonlinear functional equation, 1443 in one independent variable, 1428 power series solution, 922 nonlinear heat equation, 1301, 1318 in radial symmetric case, 1307 nonlinear integral equations, 856 with constant integration limits, 856, 863 1488 INDEX nonlinear integral equations (continued) with degenerate kernels, 863 with variable integration limit, 856 nonlinear Klein–Gordon equation, 702, 1315 nonlinear mathematical physics equations, 1301 nonlinear operator, 693, 695 nonlinear ordinary differential equations, 453–472, 488–513, 524–527 systems, 542–550 nonlinear partial differential equations, 570–584, 653–779, 1301–1335 systems, 780–798, 1343–1382 nonlinear programming, 1027 problem, 1027 nonlinear Schr ¨ odinger equation, 759, 1307 nonlinear systems of first-order ODEs, 1239 of ODEs, 542–550, 1239, 1244 of PDEs, 780–798, 1343–1382 of PDEs of general form, 1374–1384 of second-order ODEs, 1240 of two first-order PDEs, 1337 of two second-order PDEs, 1343 nonlinear Urysohn integral equation, 856 nonlinear Volterra integral equation, 856, 858, 860, 861 nonlinear wave equations, 723, 1312–1317 nonnegative Hermitian operator, 207 nonnegative quadratic form, 216 nonnegative symmetric operator, 207 nonorthogonal polynomials, 988 nonparametric hypothesis, 1094 nonpositive Hermitian operator, 207 nonpositive quadratic form, 216 nonpositive symmetric operator, 207 nonseparated by straight line points, 97 nonsingular extremal, 995 nonsingular matrix, 171 nonuniform convergence of functions, 249 norm, 314, 840 of element, 193 of linear operator, 204 normal, 91, 252, 368, 380, 389 positive sense, 369 principal, 381 to quadric, 152 to second-order curve, 111 to surface, 153 normal curvature, 392 normal curvature vector, 392 normal direction, 389 normal distribution, 1053 normal form, 473 of matrix, 181 normal fundamental system of solutions of homogeneous system, 198 normal hyperbolic equation, 590 normal matrix, 171 normal operator, 207 normal plane, 380, 382 normal representation, of quadratic form, 218 normal section, 392 normal subgroup, 227 normal system of method of least squares, 850 normal vector, 124 normalized cross-correlation function, 1073 normalized equation of plane, 126 of straight line, 91 normed Euclidean space, 193 normed unitary space, 195 noteworthy limit, second, 241 null hypothesis, 1094 null matrix, 167 null subspace, 190 number absolute value, 8 Bernoulli, 938 complex, 399 complex conjugate, 399 composite, 3 e, base of natural logarithms, xxxi, 9, 22, 23 Euler, 939 even, 3 integer, 3 irrational, 4, 5 modulus, 8 natural, 3 odd, 3 of real roots of polynomial, 165 “pi” (π), xxxii prime, 3 pure imaginary, 399 rational, 4, 5 real, 3, 4 number axis, 77 numerical eccentricity of ellipse, 98 of hyperbola, 102 numerical integration of differential equations, 471 numerical sequence, 237 generating function, 1047 numerical series, 337, 1118 numerical sum, 1113 O obelisk, 63, 65 oblate spheroid, 144 oblique asymptote, 242 oblique Cartesian coordinate system, 79 obtuse triangle, 43 octant, 114 odd function, 236 odd number, 3 odd permutation, 178 ODE, see ordinary differential equation one-dimensional Klein–Gordon equation, 618 one-dimensional stochastic process, 1072 one-parameter family of curves, 377 one-parameter Lie group of point transformations, 730 INDEX 1489 one-parameter solution, 724 one-parameter transformation, 716 one-sheeted hyperboloid, 144 imaginary semiaxis, 144 one-sided derivative, 252 one-sided limits, 240 one-soliton solution, 765 one-to-one mapping, 226 open interval, 235 open model of transportation problem, 1020 operations with asymptotic series, 364 operator, 204 adjoint, 206 bounded, 204 characteristic value, 209 differential, linear, 591, 592 differential, total, 740, 742 eigenvalue, 209 eigenvector, 209 Hermitian, 206, 207 Hermitian negative, 207 Hermitian nonnegative, 207 Hermitian nonpositive, 207 Hermitian positive, 207 identity, 204 infinitesimal, 716, 717, 730 invariant, 717, 731 inverse, 205 Laplace, xxxii, 272 linear, 204 linear invertible, 205 normal, 207 opposite, 205 orthogonal, 207 reducible, 205 root, 207 second prolongation, 720 self-adjoint, 206 skew-Hermitian, 206 skew-symmetric, 207 symmetric, 207 symmetric negative, 207 symmetric nonnegative, 207 symmetric nonpositive, 207 symmetric positive, 207 transpose, 207 unitary, 206 zero, 204 operators commutator, 755 differential, of field theory, 272 orthogonality condition, 207 opposite matrix, 169 opposite of real number, 6 optical property of ellipse, 100 of hyperbola, 103 of parabola, 105, 106 optimal solution, 1012 optimal strategy, 1024 optimality principle, Bellman, 1028 order of associated element, 213 of branch point, 414 of homogeneity, 1441 of pole, 413 of zero of function, 413 ordinary differential equation Abel, of first kind, 462 Abel, of first kind, canonical form, 463 Abel, of second kind, 464 Abel, of second kind, canonical form, 465 Airy, 955 asymptotic solutions, 477 autonomous, 489, 525, 1207, 1223, 1225 Bernoulli, 458, 1207 Bessel, 1216 Bessel, modified, 1216 boundary value problems, 480 Clairaut, 467 Darboux, 458 degenerate hypergeometric, 1215 Duffing, 501 Emden–Fowler, 1223 Ermakov, 1224 Euler, 473, 520, 1216 exact, first-order, 458 exact, second-order, 491 first boundary value problem, 480 Gaussian hypergeometric, 960, 1218 Green function, 481 homogeneous, first-order, 456 homogeneous, first-order, generalized, 456 hypergeometric, 960, 1218 hypergeometric, degenerate, 1215 Lagrange, 467 Laplace, 520 Legendre, 1217–1219 Lienard, 1225 linear, 472, 518, 1213 linear, constant-coefficient, 473, 514, 1213 linear, first-order, 457, 1207 linear, homogeneous, 472, 1212 linear, nonhomogeneous, 473 linear, second-order, 472, 1212 Mathieu, 980, 1221 Mathieu, modified, 982, 1220 method of regular expansion in small parameter, 470 method of Taylor series expansion in independent variable, 470 mixed boundary value problem, 480 Riccati, 460 Riccati, general, 461, 462, 1209, 1210 Riccati, special, 1208 second boundary value problem, 480 second-order, linear, 467 second-order, nonlinear, 488 self-adjoint form, 481 separable, 456, 1207 separated, 456 third boundary value problem, 480 Whittaker, 960 1490 INDEX ordinary differential equations, 453–550, 1207–1228 admitting reduction of order, 489, 525 asymptotic solutions, 522 boundary value problems, 480, 481, 512 Cauchy problem, 488, 524 existence and uniqueness theorem, 488 first-order, 453–472, 1207–1212 homogeneous, 490, 526 homogeneous, generalized, 490, 526 linear, 457, 472–488, 514–524, 1212–1222 linear, homogeneous, 514, 518 linear, nonhomogeneous, 515, 519 linear, of arbitrary order, 514 linear, with constant coefficients, 514 nonlinear, 453–472, 488–513, 524–527, 1223–1228 not solved for derivative, 465 numerical methods, 511 Painlev ´ e, 495 second-order, 472–514, 1212–1228 solved for derivative, 488 systems, nonlinear, 542–550 ordinate axis, 78 ordinate of point, 79, 115 oriented area, 83 oriented surface, 333 origin in space, 114 of affine coordinate system, 190 on axis, 77 on plane, 78 orthocenter of triangle, 46 orthogonal complement of subspace, 194 orthogonal coordinates, 1194 orthogonal curvilinear systems of coordinate, 1195 orthogonal elements, 193 of unitary space, 195 orthogonal group, 229 orthogonal matrix, 170 orthogonal operator, 207 orthogonal polynomials, 982 orthogonal projection, 197 orthogonal trajectory, 379 orthogonal transformation of matrix, 182 orthogonality condition for operators, 207 orthogonalization, Gram–Schmidt, 194 orthonormal basis in finite-dimensional Euclidean space, 193 in finite-dimensional unitary space, 195 osculating circle, 375 osculating plane, 381 osculation, point, 372 Ostrogradsky–Gauss theorem, 334 overdetermined systems of PDEs, 770, 771 P Pad ´ e approximants, 493 Painlev ´ e equations, 495 Painlev ´ e test for nonlinear PDEs, 748 Painlev ´ e transcendent, 495 fifth, 499 first, 496 fourth, 498 second, 496 sixth, 499 third, 497 Painlev ´ e transcendental functions, 495 pair, Lax, 755 parabola, 104 canonical equation, 104 diameters, 106 directrix, 105 focal distance, 105 focal properties, 105 focal radius, 105 focus, 105 focus-directrix property, 105 in polar coordinates, 107 optical property, 105, 106 vertex, 105 with vertical axis, 106 parabolic coordinates, 1203 parabolic cylinder coordinates, 1203 parabolic cylinder function, 967 parabolic equation, 585, 590 boundary value problems, 645 canonical form, 586 Cauchy problem, 615 in narrow sense, 590 linear, 586, 615–616, 618–622, 634, 635, 646, 648, 1267–1278 nonlinear, 653, 654, 1301–1311 parabolic point, 394 parabolic type of second-order curves, 108 paraboloid, 145, 224 elliptic, 145 hyperbolic, 146 of revolution, 146 paraboloidal cylinder, 225 parallel lines, 59 parallel planes, 59 parallel translation, 220 of axes, 79 parallel vectors, 113 parallelepiped, 61, 62 rectangular, 62 parallelogram, 52 parallelogram rule, 113 parameter of ellipse, 98 of hyperbola, 102 of integral equation, 829 of plane, polar, 127 variable, 87 parametric curves, 387 parametric equations of circle, 98 of curve, 87 of curve on plane, 87 of plane, 125 INDEX 1491 parametric equations (continued) of straight line in space, 131 of straight line on plane, 91 parametric hypothesis, 1094 parametric method for defining curve, 123 for defining surface, 123 parametrized surface, 387 Parseval identity, 361 part of Laurent series, 412 partial derivative, 264 partial differential equation biharmonic, 1297 biharmonic, homogeneous, 1297 biharmonic, nonhomogeneous, 1298 Born–Infeld, 659 Boussinesq, 709, 1332 Burgers, 664, 666, 751, 1307 Burgers, radial symmetric case, 1307 Burgers–Korteweg–de Vries, generalized, 709 Clairaut, 1262 continuity, 784 diffusion, linear, see heat equation, linear diffusion, nonlinear, with cubic source, 753 elliptic, 585, 588, 590, 1284 evolution, second-order, 739 first-order, quasilinear, 553 Fisher, 1301 Fitzhugh–Nagumo, 1302 Hamilton–Jacobi, 576 heat, linear, 585, 1267 heat, linear, nonhomogeneous, 1268 heat, nonlinear, 1318 heat, stationary anisotropic, 1323 heat, stationary, with nonlinear source, 1320 heat, with axial symmetry, 1270 heat, with central symmetry, 1272 Helmholtz, 1289 Helmholtz, three-dimensional, 600 Helmholtz, two-dimensional, 599 hyperbolic, 585, 587, 590 hyperbolic, first canonical form, 587 hyperbolic, normal, 590 hyperbolic, second canonical form, 587 Jacobi, 998, 1004, 1007 Khokhlov–Zabolotskaya, 658 Khokhlov–Zabolotskaya, stationary, 1322 Klein–Gordon, 618, 625, 1280 Klein–Gordon, linear, 625 Klein–Gordon, nonlinear, 702, 1315 Klein–Gordon, two-dimensional, 618 Korteweg–de Vries, 752, 756, 759, 1327 Korteweg–de Vries, cylindrical, 1328 Laplace, three-dimensional, 643 Laplace, two-dimensional, 585, 633, 1284 linear, constant-coefficient, 588 Liouville, 664 model, of gas dynamics, 1254 Monge–Amp ` ere, 1326 Monge–Amp ` ere, homogeneous, 668, 1326 Monge–Amp ` ere, nonhomogeneous, 1326 partial differential equation (continued) Newell–Whitehead, 733 nonlinear, fi rst-order, 570 of diffusion boundary layer, 1276 of gas dynamics, model, 558 of minimal surfaces, 770 of steady transonic gas flow, 1324 of thermal boundary layer, 1276 of transonic gas flow, 1324 of vibration of string, 1278 parabolic, 585, 586, 590 parabolic, in narrow sense, 590 Poisson, 1287 Poisson, three-dimensional, 599 Poisson, two-dimensional, 599 Schr ¨ odinger, linear, 1276 Schr ¨ odinger, nonlinear, 759, 1309 Schr ¨ odinger, of general form, 1311 Schr ¨ odinger, with cubic nonlinearity, 1309 Schr ¨ odinger, with power-law nonlinearity, 1310 second-order, nonlinear, 653 second-order, quasilinear, 654 second-order, semilinear, 653 sine-Gordon, 1314 sinh-Gordon, 1314 telegraph, 1284 telegraph, generalized, 626 ultrahyperbolic, 590 vibration, with axial symmetry, 1282 wave, 585, 1278 wave, nonlinear, 723, 1312–1317 wave, three-dimensional, 618 wave, two-dimensional, 618 partial differential equations first-order, 553, 1247 first-order, linear, 553, 1247 first-order, nonlinear, 570, 1258 first-order, quasilinear, 553, 1252 fourth-order, linear, 1294–1298 fourth-order, nonlinear, 1332–1334 nonlinear, 653–769, 1301–1335 nonlinear systems, 770–798, 1337–1341, 1343–1382 second-order, linear, 585–650, 1267–1299 second-order, linear, classification, 585 second-order, nonlinear, classification, 653, 654 third-order, nonlinear, 1327–1331 partial fraction decomposition of rational function, 276 partial increments, 264 of function, 264 partial limit of sequence, 240 partial sum of series, 337 particular solution, 599 partition diameter, 317 path integral, 329 payoff matrix, 1024 saddle point, 1025 PDE, see partial differential equation Peano existence theorem for system of ODEs, 543 Pearson first skewness coefficient, 1045 1492 INDEX Pearson test, 1100 penalty function, 1027 penalty function method, 1027 pencil, 92 center, 92 of planes, 131 of straight lines, 92 percentage, 7 simple, 7 percentile, 1045 periodic function, 236 permutation, 178 even, 178 odd, 178 permutation groups, 226 permutations, 1033, 1034 with repetitions, 1034 perpendicular bisector, 46 perturbation methods, 499 Peterson–Codazzi formulas, 396, 397 Pexider equation, 925, 1438 Pfaffian equation, 772 integrating factor, 773 phase plane, 546 phase variable, 546, 1010 physical meaning of line integral of second kind, 331 Picard method, 469 pivot column, 1016 pivot entry, 1016 pivot row, 1016 plane, 124 complex, 401 condition for planes to be parallel, 138 condition for planes to be perpendicular, 138 condition for planes to coincide, 138 coordinate axes, 78 coordinate system, 78 curve, 84 deviation of point from plane, 141 general (complete) equation, 124 incomplete equation, 124 intercept equation, 125 normal, 380, 382 normalized equation, 126 osculating, 381 parametric equation, 125 phase, 546 principal, 151 rectifying, 382 tangent, 268, 381, 389 tangent to surface, 152 plane curves, 367 plane geometry, 43 plane in space, 124 plane polygon, 51 plane triangle, 43 planes in space, 59 parallel, 59 player, 1024 Pochhammer symbol, 938 Poincar ´ e type series, asymptotic, 363 point applicate, 115 asymptotic, 373 at infinity, 402 at infinity, neighborhood, 402 boundary of domain, 401 characteristic, 377 circular, 393 conjugate, 998, 1000, 1004, 1007 coordinates, 79, 115 derivative of function, 402 deviation from line, 93 deviation from plane, 141 double, 371 elliptic, 394 Feuerbach, 49 fixed, 908 Gergonne, 47 hyperbolic, 394, 536 incenter, 46 isolated, 371 movable critical, 495 Nagel, 47 of affine space, 189 of extremum, 257, 269 of intersection of straight lines, 95 of local minimum, 269 of osculation, 372 of rectification, 382 of self-intersection, 372 of tangency, 56 on complex plane, 402 on plane, 89 parabolic, 394 ramification, 414 regular, 367, 379, 387 saddle, 394 saddle, of payoff matrix, 1025 singular, 371, 391 singular of generalized solution, 580 spiral stable, 537 spiral unstable, 537 stationary, 220, 269, 545 termination, 373 transcendental branch, 415 transition, 477 triple, 371 umbilical, 393 point estimators, 1088 point transformation, 455 for ODEs, 455 for PDEs, 655 point-slope equation of straight line, 89 points collocation, 848 nonseparated by straight line, 97 separated by straight line, 97 pointwise convergence of functional series, 348 of functions, 249 INDEX 1493 Poisson distribution, 1050 Poisson equation, 1287 Poisson formula for Bessel function, 949 in probability theory, 1038 Poisson integral, 428, 1287 for half-plane, 429 Poisson process, 1076 Poisson theorem, 428, 1070 polar, 112 polar angle, 80 polar axis, 80 polar bilinear form, 216 polar coordinate system, 80 polar coordinates, 80, 1198 polar decomposition of linear operator, 208 of matrices, 173 polar kernel, 801, 825 polar parameter of plane, 127 polar radius, 80 pole, 80, 413 order, 413 simple, 413 polygon, 51 center, 55 circumscribed about circle, 51 convex, 51 distribution, 1085 edge, 51 external angle, 51 frequency, 1085 inscribedincircle,51 interior angle, 51 plane, 51 relative frequency, 1085 side, 51 simple, 51 vertex, 51 polyhedra, 61 polyhedral angle, 60 edge, 61 face, 60 vertex, 60 polyhedral angles congruent, 61 symmetric, 61 polyhedron, 61 convex, 61 edge, 61 face, 61 vertex, 61 polynomial, 155 annihilating, 186 Bernoulli, 891, 988 characteristic, of linear operator, 211 characteristic, of matrix, 185 Chebyshev of first kind, 983 Chebyshev of second kind, 984 derivative, 157 Euler, 989 polynomial (continued) expansion in powers of linear binomial, 156 Gegenbauer, 987 Hermite, xxxi, 957, 958, 985 interpolation, 186 Jacobi, 986 Laguerre, 982, 957 Laguerre generalized, 983 Legendre, xxxi, 962, 1217 minimal, 186 number of real roots, 165 remainder theorem, 156 ultraspherical, 987 with matrix argument, 172 with matrix coefficients, 172 polynomials difference, 155 division, 155 equal, 155 multiplication, 155 orthogonal, 982 nonorthogonal, 988 product, 155 sum, 155 Pontryagin function, 1011 Pontryagin maximum principle, 1011 population, 1081 expectation, 1087 variance, 1087 position vector of point, 115 positive definite kernel, 837 positive definite quadratic form, 216 positive definite quadratic functional, 993 positive Hermitian operator, 207 positive index of inertia of quadratic form, 218 positive integer power of complex number, 399 of matrix, 172 positive normal direction, 389 positive operator, 207 positive sense, 367 of boundary, 402 of normal, 369 of tangent, 369 positive sequence, 238 positive symmetric operator, 207 Post–Widder formula, 441 potential, 331, 760, 1021 of gravitational field, 328 of vector field, 331 potential-type integrals, 291 power, test, 1096 power function, 19 graphs, 19 properties, 19 power Laplace integral, 294 multiple, 328 power series, 350, 1120 Abel theorem, 350 addition, 354 convergence radius, 350 division, 354 . 1327–1331 partial fraction decomposition of rational function, 276 partial increments, 264 of function, 264 partial limit of sequence, 240 partial sum of series, 337 particular solution, 599 partition. 1307 nonlinear systems of first-order ODEs, 1239 of ODEs, 542–550, 1239, 1244 of PDEs, 780–798, 1343–1382 of PDEs of general form, 1374–1384 of second-order ODEs, 1240 of two first-order PDEs, 1337 of two second-order. 495 Nagel, 47 of affine space, 189 of extremum, 257, 269 of intersection of straight lines, 95 of local minimum, 269 of osculation, 372 of rectification, 382 of self-intersection, 372 of tangency,