1494 INDEX power series (continued) interval of convergence, 351 Maclaurin, 350 multiplication, 354 properties, 350 solution of nonlinear functional equations, 922 subtraction, 354 summation formulas, 357 Taylor, 350 powers, 8 of hyperbolic functions, 37 of linear operators, 205 of matrices, 172 of trigonometric functions, 28 price, upper of game, 1025 primal linear programming problem, 1017 prime number, 3 primitive function, 273 principal axis, 109, 151, 152 of hyperbola, 101 of second-order curve, 109 principal diameter of ellipse, 100 of hyperbola, 104 of parabola, 106 principal directions of second-order curve, 109 principal normal, 381 principal normal sections, 393 principal part of Laurent series, 412 principal plane, 151 principal value, Cauchy, of singular integral, 311 principle boundary correspondence, 420 Duhamel first, 646 Duhamel second, 648 maximin, 1025 maximum modulus, 403 minimax, 1025 of linear superposition, for PDEs, 594 Rayleigh–Ritz, 484, 513 Riemann localization, 358 Schwarz analytic continuation, 426 superposition, for homogeneous system of linear ODEs, 528, 539 superposition, for linear functional equations, 880, 883 superposition, for linear PDEs, 594 superposition, for nonhomogeneous linear ODE, 519 superposition, for nonhomogeneous system of linear ODEs, 541 prism, 61 base face, 61 joining face, 61 right, 62 truncated, 62 Privalov boundary value problem, 432 probabilistic models, 1031 probability, 1032–1034 classical definition, 1033 conditional, 1035 probability (continued) convergence, 1068 geometric, 1034 multiplication theorem, 1035 of event, 1032 transition, 1075 probability density function, 1040 probability element, 1040, 1059, 1065 probability function joint, 1059, 1064 marginal, 1059 probability integral, 939, 1265 probability multiplication theorem, 1035 probability space, 1032 discrete, 1033 probability theory, 1031 probability-generating function of random variable, 1047 problem Banach, 1038 Bolza, 1001 boundary value, first, 480, 593, 627 boundary value, for difference equations, 879 boundary value, for elliptic equations, 631 boundary value, for hyperbolic equations, 623 boundary value, for ODEs, 480 boundary value, for parabolic equations, 645 boundary value, Hilbert–Privalov, 432 boundary value, mixed, 480, 593, 627 boundary value, nonhomogeneous, 620, 624 boundary value, second, 480, 593, 627 boundary value, third, 480, 593, 627 Cauchy, for difference equations, 889 Cauchy, for first-order ODEs, 454 Cauchy, for first-order PDEs, 556, 576 Cauchy, for Hamilton–Jacobi equation, 577 Cauchy, for hyperbolic equations, 617 Cauchy, for parabolic equations, 615 Cauchy, for PDEs, 591, 592, 785, 1308 Cauchy, for second-order ODEs, 488 Cauchy, for system of PDEs, 785 Cauchy, generalized, for PDEs, 627 Dido, 1003 diet, 1012 direct scattering, 760 Dirichlet, 427 Dirichlet, for elliptic equations, 593, 1330 Dirichlet, generalized, 428 dual linear programming, 1017 eigenvalue, 604 fi rst boundary value, 593 general mathematical programming, 1012 Goursat, 592, 629 initial value, 580, 785 inverse scattering, 760 isoperimetric, 1002 Lagrange, 1008 linear programming, 1012 Neumann, 429 Neumann, for elliptic equations, 593, 1328 nonlinear programming, 1027 of propagation of signal, 566 INDEX 1495 problem (continued) primal linear programming, 1017 Privalov boundary value, 432 product mix, 1012 programming linear, 1012 Riemann, 785 simplest, of calculus of variations, 993 Sturm–Liouville, 482, 621, 622, 625, 626, 633 terminal value, 580, 583, 584 third boundary value, 593 transportation, 1019 transportation balanced, 1020 two-body, 1258 with higher derivatives, 1006 problems for elliptic equations, 637, 643, 644 for hyperbolic equations, 607, 636 for parabolic equations, 605, 634 of mathematical physics, 590 of mathematical statistics, 1081 process Bernoulli, 1037 birth-death, 1077 iteration, 858 Markov, 1075 optimal, in strong sense, 1010 point simplest, 1076 Poisson, 1076 pure birth, 1077 pure death, 1077 stationary, 1075 stochastic, 1071, 1073, 1074 stochastic multidimensional, 1072 stochastic one-dimensional, 1072 stochastic stationary in narrow sense, 1074 stochastic stationary in wide sense, 1074 stochastic vector, 1072 stochastic with continuous time, 1072 stochastic with discrete time, 1072 Wiener, 1086 product inner, 192, 314, 840 Kronecker of matrices, 175 mix problem, 1012 of complex numbers, 399 of linear operators, 203 of linear transformations, 227 of matrices, 157 of matrix by scalar, 157 of polynomials, 155 of real numbers, 6 of scalar and linear operator, 203 of scalar and vector, 187 of subsets of group, 227 of vector by number, 113 scalar, 192 scalar of vectors, 120 scalar triple of vectors, 123 triple cross of vectors, 122 products of hyperbolic functions, 37 of trigonometric functions, 28 programming convex, 1027 dynamic, 1027 geometric, 1027 quadratic, 1027 progression, 1113, 1118 arithmetic, 11 geometric, 11 projection, 212 of vector onto axis, 114 on segment, 81 orthogonal, 197 projection methods, 508 projections, theorem, 44 prolate spheroid, 144 prolongation of group generator, 732 of operator, second, 720 proof by contradiction, 15 proof by counterexample, 17 proper orthogonal transformations, 229 proper subspace, 190 properties of limits, 240 property Markov, 1075 minimax, 210 optical of ellipse, 100 optical of hyperbola, 103 optical of parabola, 106 stability, 1055 proportions, 7 psi function, 944 Ptolemy theorem, 52 Puiseux power series, 415 pure birth process, 1077 pure imaginary complex number, 399 pyramid, 63 altitude, 63 apex, 63 apothem, 64 base, 63 frustum, 64 lateral face, 63 triangular, 63 Pythagorean theorem, 50, 193 Q quadrant, 79 quadratic equation, 158 quadratic form, 216, 840 alternating, 216 canonical representation, 217 characteristic, 150 characteristic equation, 150 degenerate, 216 extremal properties, 220 in Euclidean space, 219 in finite-dimensional linear space, 216 index of inertia, 218 negative definite, 216 1496 INDEX quadratic form (continued) negative index of inertia, 218 nondegenerate, 216 nonnegative, 216 nonpositive, 216 normal representation, 218 positive definite, 216 positive index of inertia, 218 rank, 216 second of surface, 392 signature, 218 quadratic functional, 992 positive definite, 993 strongly positive, 993 quadratic mean, 13 of functions, 297 quadratic nonlinearity, 865 quadratic programming, 1027 quadrature formula, 808, 860 quadrature method, for integral equations, 852, 861, 870 quadrature nodes, 808 quadric canonical equations, 143 canonical form, 143, 148 classification, 148 general equation, 148 general theory, 148 invariant, 148 semi-invariant, 148 quadric surfaces, 143 quadrilaterals, 52 quartile, 1045 quasidiagonal canonical form of matrix, 182 quasilinear equations in conservative form, 567 quasilinear partial differential equations, 553 quotient, 155 of complex numbers, 399 quotient group, 227 R Raabe convergence criterion for series, 339 radian, 24 radiation conditions, 1290, 1291 radical, 8 radius focal left, 99, 102 of convergence, 351 of convergence, of power series, 411 of curvature, 376, 384 of torsion, 385 polar, 80 ramification point, 414 random event, 1031, 1032 random sampling with replacement, 1034 without replacement, 1033 random sequence, 1072 random variable, 1039, 1057 bivariate, 1060 central moment, 1065 random variable (continued) continuous, 1040, 1042, 1064 continuous bivariate, 1059 discrete, 1039, 1042, 1064 discrete bivariate, 1058 expected value, 1042 initial moment, 1060, 1065 symmetric, 1041 symmetric about expected value, 1041 variance, 1044 random variables correlated, 1061 independent, 1060, 1067 joint distribution function, 1057 uncorrelated, 1061 random vector, 1057 range of function, 236 of linear operator, 205 of mapping, 226 sample, 1082 rank of bilinear form, 215 of linear operator, 206 of matrix, 178 of quadratic form, 216 Rankine–Hugoniot jump conditions, 561, 568, 790, 791 rarefaction wave, 559, 785, 1253 rate false negative, 1096 false positive, 1095 ratio common, 11 contraction, 403 dilatation, 403 likelihood, 1097 Romanovskii, 1101 Student, 1090 rational function, 276 rational numbers, 4, 5 rational polynomial function, 276 Rayleigh–Ritz principle, 484, 513 real axis, 235, 401 of hyperbola, 101 real Euclidean space, 192 real linear space, 188 real matrix, 171 real number, 3, 4 multiplicative inverse, 6 reciprocal, 6 real part, 399 real semiaxes of hyperbola, 102 of one-sheeted hyperboloid, 144 of two-sheeted hyperboloid, 144 real-valued function of real variable, 236 reciprocal (algebraic) equation, 160, 161 reciprocal functional equation, 919 reciprocal of real number, 6 rectangle rule, 315, 808 for computation of integrals, 315 INDEX 1497 rectangular Cartesian coordinate system, 78, 114 rectangular Cartesian coordinates on plane, 78 rectangular matrix, minor, 177 rectangular parallelepiped, 62 rectangular spherical triangle, 72 rectification, point, 382 rectifying plane, 382 reduced cost, 1015 reducible operator, 205 reducible representation, 230 reduction of central second-order curves to canonical form, 110 of noncentral second-order curves to canonical form, 110 of quadratic form to sum of squares, 219 of square matrix to diagonal form, 184 of square matrix to triangular form, 184 reduction formulas, 1145 for trigonometric functions, 27 reflectance, 760 reflection coefficient, 760 reflection factor, 760 regime blow-up, 1303, 1304 critical, 1095 feasible, 1012 regression, 1063, 1066 mean-square, 1063, 1066 regression analysis, 1105 regression coefficient, 1063 regression function, 1063, 1066 regula falsi method, 262 regular arc, 379 regular extremal, 995 regular part of Laurent series, 412 regular point, 367, 379, 387 of extremal, 995 of generalized solution, 580 of plane curve, 367 of space curve, 379 regular polygon, 55 center, 55 regular polyhedron, 65 regular prism, 62 regular pyramid, 64 regular triangle, 43, 51 regular value, 829, 834 regularization methods for integral equations, 827 relation Parseval for Fourier cosine transform, 445 Parseval for Fourier sine transform, 446 Parseval for Hankel transform, 447 relative extrema, 991 relative frequencies, 1083 relative frequency histogram, 1086 relative frequency of event, 1083 relative frequency polygon, 1085 remainder in Taylor formula in form of Cauchy, 257 remainder (continued) in Taylor formula in form of Lagrange, 257 in Taylor formula in form of Peano, 257 in Taylor formula in form of Schl ¨ omilch and Roche, 257 in Taylor formula in integral form, 257 of functional series, 349 of series, 337 removable singularity, 413 renewal equation, 1395 representation dimension, 230 invariant, 230 irreducible, 231 of group, completely reducible, 231 of quadratic form canonical, 217 of vector in terms of basis, 189 reducible, 230 triangular, 201 representation basis, 230 representation space, 230 residual, 847, 1066 residual standard deviation, 1063 residual variance, 1066, 1108 residue, 415, 435 logarithmic, 416 resolvent, 810, 823, 829, 833 of symmetric kernel, 839 resonance, Painlev ´ e test, 749 rhombus, 53 Riccati difference equation, 875, 918, 1428, 1429 Riccati differential equation, 460 general, 461, 462, 1209, 1210 polynomial solutions, 460 reduction to canonical form, 460 reduction to second-order linear equation, 461 special, 1208 Riemann function, 628, 629, 783 Riemann integral, 286 Riemann invariant, 783, 786, 787 Riemann localization principle, 358 Riemann problem, 785 for system of PDEs, 785 Riemann property, 294 Riemann sum, 286 Riemann theorem, 419 Riemann wave, 782 Riemann–Schwarz theorem, 426 Riemannian space, 395 right arm of hyperbola, 101 right circular cone, 68, 145 right circular cylinder, 66 right coset, 227 right cylinder, 66 right directrix of ellipse, 99 of hyperbola, 102 right ellipsoid, 145 right focal radius, 99, 102 right focus of ellipse, 99 right focus of hyperbola, 102 1498 INDEX right half-plane, 79 right inverse matrix, 171 right prism, 62 right rectangular Cartesian coordinate system, 79 right triangle, 43, 50 leg, 50 right-angled triangles, 43, 50 right-hand limit, 240 right-hand side of integral equation, 810 right-handed trihedral, 121 Rolle theorem, 254 Romanovskii ratio, 1101 Romanovskii statistic, 1101 Romanovskii test, 1101 root of complex number, 399 of equation, 162 of multiplicity m, 162 of operator, 207 root mean square, 13 root-finding algorithms for continuous functions, 260 root-mean-square deviation, 1045 rotation of axes, 79 rotation transformation, 656 Rouch ´ e theorem, 413 round cylinder, 66 segment, 67 Routh–Hurwitz criterion, 167 row, pivot, 1016 row expansion of determinant, 176 row vectors, 167 linear combination, 171 linearly dependent, 171 linearly independent, 171 rows, basic, 165 rule Cramer, 199 Lagrange multiplier, 1002 of closing chain of vectors, 113 parallelogram, 113 rectangle, 315, 808 sigma, 1054 Simpson, 315, 808 trapezoidal, 315, 808 triangle, 113, 176 Whitham, “equal areas”, 562 ruled surface, 146 rules, Neper, 75 rulings, 146 of ruled surfaces, 146 Runge–Kutta method, 472, 512 S saddle, 536 saddle point, 394 of payoff matrix, 1025 sample, 1082 asymmetry coefficient, 1088 central moment, 1087 correlation, 1104 sample (continued) distribution function, 1084 excess coefficient, 1088 function, 1072 mean, 1087 mean square deviation, 1088 moment, 1087 range, 1082 size, 1082 space, 1031 variance, 1088 variance adjusted, 1088 sampling random with replacement, 1034 random without replacement, 1033 Sapogov convergence criterion for series, 340 Sarrus scheme, 176 scalar, 187 scalar product, 192 of vectors, 120 scalar square, 120 scalar triple product of vectors, 123 scale segment, 77 scalene ellipsoid, 144 scalene triangle, 43 scaling transformation, 656, 676, 923 scattering data, 761 scheme Bernoulli, 1037 Cole–Kevorkian, 504 Horner, 156 Sarrus, 176 Van der Pol–Krylov–Bogolyubov, 503 schlicht mapping, 402 Schl ¨ omilch equation, 1400 Schr ¨ oder–Koenigs functional equation, 911 Schr ¨ odinger equation linear, 1276 nonlinear, of general form, 1311 with cubic nonlinearity, 1309 with power-law nonlinearity, 1310 Schwarz analytic continuation principle, 426 Schwarz integral, 428 for half-plane, 429 Schwarz–Christoffel theorem, 427 secant, xxxi, 24 second boundary value problem, 480, 593 for elliptic equations, 633 for hyperbolic equations, 627 for PDEs, 593 second convergence criterion, 338 second differential, 266 second duality theorem, 1018 second group of derivation formulas, 396 second mean value theorem, 295 second noteworthy limit, 241 second Painlev ´ e transcendent, 496 second partial derivatives, 266 second prolongation of operator, 720 second quadratic form of surface, 392 second variation of functional, 992, 993 INDEX 1499 second-order constant coefficient linear ODE, 473, 1213 second-order curve, 107 canonical form, 107 center, 109 central, 109 characteristic equation, 108 diameter, 108 elliptic type, 108 general equation, 107 hyperbolic type, 108 invariant, 107 on plane, 97 normal, 111 semi-invariant, 108 tangent, 111 vertex, 109 second-order difference equations linear homogeneous, 877, 1421 linear nonhomogeneous, 879, 880, 1421, 1422 nonlinear, 1436 second-order differential constraint, 744 second-order evolution equation, 739 second-order hypersurface, 220 center, 222 centeral, 222 equation, 223 general equation, 220, 222 noncenteral, 224 second-order ordinary differential equations linear, 472–488, 1212–1222 linear, general solution, 472 nonlinear, 488–514, 1223–1228 second-order partial differential equations linear, 585–650, 1267–1298 nonlinear, 653, 1301–1334 quasilinear, 654 semilinear, 653 second-order variation of functional, 993 secondary diagonal, 168 section, normal, 392 sections, principal normal, 393 sector, 58 circular, 58 spherical, 69 secular terms, 502 segment, 58, 235 directed, 77 direction cosine, 116 length, 77 of round cylinder, 67 on axis, 77 scale, 77 spherical, 69 value, 77 self-adjoint matrix, 168 self-adjoint operator, 206 self-intersection, point, 372 self-similar solution, 669, 1442 continuous, 781 exponential, 675 self-similar solution (continued) generalized, 677 limiting, 675 semi-invariant for probability distribution, 1046 of quadric, 148 of second-order curve, 108 semiaxis, 826, 842 imaginary, of hyperbola, 102 imaginary, of one-sheeted hyperboloid, 144 imaginary, of two-sheeted hyperboloid, 144 of ellipsoid, 143 of hypersurface, 223 semilinear partial differential equations, 653 semimajor axis of ellipse, 98 semiminor axis of ellipse, 98 separable equation, 1207 separable solution, 596 additive, 596 multiplicative, 596 separated by straight line points, 97 separated equation, 456 separation constant, 678 separation of variables, 602 complex, 679 functional, 697 generalized, 681 incomplete, 597, 616, 640 simple, 678 sequence arithmetic, 11 bounded, 237 convergent, 237, 238 decreasing, 238 generic term, 237 geometric, 11 increasing, 238 infinitely large, 239 limit, 237 lower limit, 240 monotone, 238 numerical, 237 random, 1072 upper limit, 240 sequential analysis, 1097 series, 337 Abel convergence criteria, 341 Abel theorem, 343 Abel transformation, 344 absolute convergence, 341 arbitrary, 341 arithmetic, 11 asymptotic, 363 Cauchy theorem, 343 conditional convergence, 341 convergence domain, 348 convergent, 337 Dirichlet convergence criteria, 341 divergent, 337 Fourier, 357, 358 fractional power, 415 functional, 348 1500 INDEX series (continued) generic term, 337 geometric, 11 harmonic, 339 hypergeometric, 960, 1218 interval, 1083 interval of convergence, 351 Kummer, 956, 1215 Kummer transformation, 344 Laurent, 410, 412 Leibnitz convergence criterion, 341 Maclaurin, 353 majorant, 349 Mertens theorem, 343 Neumann, 823, 833 numerical, 337 of order statistics, 1082 of Poincar ´ e type, asymptotic, 363 power, 350 Puiseux power, 415 remainder, 337 statistical, 1083 sum, 337 summation methods, 344 Taylor, 352, 410, 411 uniformly convergent, 349 series solutions for hyperbolic equations, 607 for ODEs, 470, 475, 492 for parabolic equations, 605 series with positive terms convergence criteria, 338 divergence criteria, 338 Serret–Fr ´ enet formulas, 386 sesquilinear form in unitary space, 213 matrix, 214 set bounded, 263 countable, 3 diameter, 317 modulus, 907 submodulus, 907 unbounded, 263 shock wave, 790–793, 1255 shooting method, 512 shortest distance, 70 side of polygon, 51 sigma rule, 1054 sign function, xxxi signature of matrix, 182 of quadratic form, 218 similar matrices, 181 similarity method, 669 similarity transformation, 181 simple hypothesis, 1094 simple percentage, 7 simple pole, 413 simple polygon, 51 simple Riemann wave, 782, 783 simple separation of variables, 678 simple zero, 413 simplest point process, 1076 simplest problem of calculus of variations, 993 simplex method, 1014 simply connected domain, 409 Simpson line, 47 Simpson rule, 315, 808 for computation of integrals, 315 sine, 24, 25 sine integral, 941 sine-Gordon equation, 1314 single-step methods of second-order approxima- tion, 471 single-valued function, 236, 402 singular integral, 310, 311, 570, 571 in sense of Cauchy principal value, 430 principal value, 311 singular point, 371, 391 of generalized solution, 580 of plane curve, 371 of solutions, 494 of surface, 391 singular solutions, 455 singularity essential, 413 isolated, 413 removable, 413 sinh-Gordon equation, 756, 1314 sink, 535, 538 spiral, 537 sinusoid, sine curve, 25 sixth Painlev ´ e transcendents, 499 size sample, 1082 test, 1095 skew lines, 59 skew self-distributivity equation, 1449 skew-Hermitian matrix, 168 skew-Hermitian operator, 206 skew-symmetric bilinear form, 214 skew-symmetric integral equation, 840 skew-symmetric matrix, 168 skew-symmetric multilinear form, 215 skew-symmetric operator, 207 skewness, 1045 slope of straight line, 89 slope of tangent, 368 slope-intercept equation of straight line, 89 small circle, of a sphere, 71 small discriminant, 108 smallest value of function, 258 Smirnov test, 1099 Sokhotskii theorem, 431 solid angle, 61 solid geometry, 59 soliton, 1325 soliton solution, 765 of Korteweg–de Vries equation, 764 soliton theory, 755 . 6 of scalar and linear operator, 203 of scalar and vector, 187 of subsets of group, 227 of vector by number, 113 scalar, 192 scalar of vectors, 120 scalar triple of vectors, 123 triple cross of. 230 reduction of central second-order curves to canonical form, 110 of noncentral second-order curves to canonical form, 110 of quadratic form to sum of squares, 219 of square matrix to diagonal form,. of event, 1083 relative frequency polygon, 1085 remainder in Taylor formula in form of Cauchy, 257 remainder (continued) in Taylor formula in form of Lagrange, 257 in Taylor formula in form of