1480 INDEX inverse trigonometric functions, 30 arccosine, 32 arccotangent, 32 arcsine, 31 arctangent, 32 differentiation formulas, 34 expansion in power series, 34 integration formulas, 34 inversion, 178, 422 invertible linear operator, 205 invertible mapping, 226 invertible transformation, 267 involutive matrix, 172 involutory function, 917, 1423 irrational numbers, 4, 5 irreducible representation, 231 isolated point, 371 isolated singularity, 413 isometric correspondence, 395 isomorphic Euclidean spaces, 194 isomorphic groups, 227 isomorphic linear spaces, 189 isomorphic mapping of groups, 227 isomorphic unitary spaces, 196 isomorphism, 226, 227 isomorphisms of linear spaces, 188 isoperimetric constraints, 1002 isoperimetric problem, 1002 isosceles trapezoid, 54 isosceles triangle, 43, 50 base, 50 leg, 50 iterated integral, 319 iterated kernel, 822, 832, 838 iteration axis, 876 iteration of function, 907 iteration process, 858 J Jacobi condition, 998, 1000, 1004, 1007 strengthened, 1000, 1004, 1007 Jacobi elliptic functions, 979 Jacobi equation, 998, 1004, 1007 Jacobi method, 217 Jacobi polynomials, 986 Jacobi theorem, 180 Jacobi theta functions, 978 Jacobi–Mayer bracket, 574 Jacobian, 267, 321 Jacobian determinant, 321 Jensen equation, 1436 Jensen inequality, 15, 245, 297 joining face of prism, 61 joint distribution function, 1064 of random variables, 1057 joint probability function, 1059, 1064 Jordan cell, 182 Jordan form, of matrix, canonical, 182 Jordan lemma, 417, 436 jump conditions, 561, 568, 791 K K-transform, 449 kernel degenerate, 801, 802, 810, 811, 825, 829 difference, 801, 804, 810, 813, 825, 829, 841 Fredholm, 825, 829 iterated, 822, 832 logarithmic, 801 of integral equation, 801, 825, 829 of integral transform, 435 of linear operator, 205 polar, 801, 825 positive definite, 837 symmetric, 825, 829 with weak singularity, 801, 825, 829 Khokhlov–Zabolotskaya equation, 658 Klein–Gordon equation, 625, 1280 Kolmogorov distribution, 1086 Kolmogorov test, 1099 Kolmogorov theorem, 1070 Kontorovich–Lebedev inversion formula, 448 Kontorovich–Lebedev transform, 448, 449 Korteweg–de Vries equation, 752, 756, 759, 764, 1327 cylindrical, 1328 modified, 1328 Kronecker delta, xxxii Kronecker product of matrices, 175 Kronecker–Capelli theorem, 198 kth absolute moment, 1044 kth central moment, 1043 kth moment, 1043 kth sample central moment, 1087 kth sample moment, 1087 Kummer convergence criterion for series, 340 Kummer functions, 956 Kummer series, 956, 1215 Kummer transformation, 345, 957 Kummer–Liouville transformation, 475 kurtosis, 1045 L L 2 -norm, 314 L’Hospital rules, 254 L’Huiller equation, 74 Lagrange equation, 467 Lagrange function, 270 Lagrange method, 217 Lagrange multiplier, 270 Lagrange multiplier rule, 1002 Lagrange principle, 1008 Lagrange problem, 1008 Lagrange theorem, 254 Lagrange–Charpit method, 573 Lagrangian, 767, 993, 1002, 1006 Bolza, 1001 Noetherian symmetry, 768 Laguerre polynomials, 957, 982 generalized, 983 Lalesco–Picard equation, 1403 INDEX 1481 Lam ´ ecoefficients, 1196 Landen transformations, 974, 975 Laplace cascade method, 600 Laplace equation ODE, 520 PDE, three-dimensional, 643 PDE, two-dimensional, 633, 1282 Laplace integral, 418, 521, 963 general form, 292 special form, 291 Laplace operator, 272 Laplace theorem, 177 Laplace transform, 436, 548, 611, 804, 813, 843, 856 convolution theorem, 437 for derivatives, 611 inverse, 437 limit theorems, 440 properties, 437 solving linear problems for hyperbolic equations, 614 solving linear problems for parabolic equations, 612 tables, 1157–1163 Laplace-type multiple integral, 328 large discriminant, 108 large numbers strong law, 1070 weak law, 1069 largest value of function, 258 lateral face of pyramid, 63 latitude, 119 Laurent series, 410, 412 Lavrentiev regularization method, 827 law commutative composition, 225 composition, 225 conservation, 766 conservation at discontinuity, 561 energy conservation, 996 of conservation of mass, 784 of cosines, 44 of cosines of angles, 73 of cosines of sides, 73 of large numbers, 1069, 1070 of sines, 44, 73 of tangents, 44 weak of large numbers, 1069 laws, de Morgan, 1032 Lax condition, 792 Lax pair, 755 least common multiple, 4 least-squared error method, 511 least-squares method, 509, 1092 left arm of hyperbola, 101 left coset, 227 left directrix of ellipse, 99 of hyperbola, 102 left focal radius, 99, 102 left focus of ellipse, 99 left focus (continued) of hyperbola, 102 left half-plane, 79 left inverse matrix, 171 left rectangular Cartesian coordinate system, 79 left-hand limit, 240 leg of isosceles triangle, 50 of right triangle, 50 of trapezoid, 54 Legendre condition, 998, 999, 1004, 1007 strengthened, 999, 1004, 1007 Legendre equation, 1217–1219 Legendre function, 962 associated, 964, 965 first kind, associated, 965 modified associated, 966 second kind, 1217 second kind, associated, 965 Legendre functions, Wronskians, 967 Legendre polynomial, xxxi, 962, 1217 Legendre theorem, 72 Legendre transformation, 469, 660 with many variables, 663 Leibnitz convergence criteria, 341 Leibnitz convergence criterion for series, 341 Leibnitz formula, 256 lemma Jordan, 417 Neyman–Pearson, 1097 length of element, 193 of segment, 77 of vector, 113 level, confidence, 1093 level line, 264, 1014 Levi-Civita symbol, 179 L ´ evy solution, 914 Lie group, 730 of point transformations, 730 Lienard equation, 1225 light rays, equation, 1259 likelihood equation, 1092 likelihood function, 1092 likelihood ratio, 1097 likelihood ratio test, 1097 limit first noteworthy, 241 left-hand, 240 lower of sequence, 240 noteworthy second, 241 of function, 240, 264 of sequence, 237 partial of sequence, 240 right-hand, 240 upper of sequence, 240 limit theorems, 1068, 1069 for Bernoulli process, 1038 for Laplace transform, 440 limiting self-similar solution, 675 Lindeberg central limit theorem, 1071 1482 INDEX line deviation of point from line, 93 Euler, 47 in space, 131 level, 264, 1014 Simpson, 47 support, 1014 tangent, 251 tangent to surface, 388 line integral first kind, 329 second kind, 330, 331 linear algebraic equation, 157 linear combination of column vectors, 171 of row vectors, 171 of vectors, 188 linear constant-coefficient partial differential equation, 588 linear difference homogeneous equation with constant coefficients, 906 linear eccentricity of ellipse, 98 of hyperbola, 102 linear equation algebraic, 157 difference, 881, 885, 889, 1421, 1425 difference, constant-coefficient, 1409, 1422, 1426 integral, 801, 810, 824, 829 ordinary differential, 472, 518, 1213 partial differential, elliptic, 592 partial differential, hyperbolic, 592 partial differential, parabolic, 591 linear equations of mathematical physics, 1267 linear first-order ODE, 457 linear form, 213 linear functional, 213 linear functional equations, 907, 1414–1421, 1438 in one independent variable, 1409 linear group, general, of dimension n, 228 linear harmonic oscillator, 1277 linear homogeneous difference equations, 904 with constant coefficients, 894, 904 with variable coefficients, 895, 901 linear homogeneous functional equations, 910 linear integral equations first kind, with constant limits of integration, 824, 1396–1400 first kind, with variable limit of integration, 801, 1385–1390 second kind, with constant limits of integra- tion, 829, 1401–1408 second kind, with variable limit of integration, 810, 1391–1395 linear nonhomogeneous difference equations, 906 with constant coefficients, 895, 899 with variable coefficients, 896, 902 linear nonhomogeneous functional equations, 912 linear operator, 204 canonical form, 213 linear operator (continued) characteristic equation, 211 determinant, 209 in matrix form, 208 injective, 205 invariant, 211 inverse, 205 invertible, 205 kernel, 205 matrix, 208 nondegenerate, 205 norm, 204 range, 205 rank, 206 trace, 209 linear operators product, 205 sum, 204 linear ordinary differential equations, 457, 472–488, 514–524, 1212–1222 canonical form, 473 linear partial differential equations, 553–650, 1267–1298 canonical form, 585–590 linear programming problem, 1012 standard form, 1013 symmetric, 1013 linear representation of group, 230 linear space, 187, 189 basis, 189 complex, 188 dimension, 189 elements, 187 infinite-dimensional, 189 linear transformation, 204 of affine space, associated, 189 linear spaces isomorphic, 189 isomorphisms, 188 linear span, 190 linear subspace, 190 invariant under nonlinear operator, 693, 695 linear superposition principle, 594 linear surface element, 390 linear systems of algebraic equations, 197 of ODEs, 528 of PDEs of general form, 1374 of two second-order PDEs, 1341 linear transformation, 656 of linear space, 204 linear transformations, product, 228 linear-fractional mappings, 422 linearly dependent column vectors, 171 linearly dependent elements, 188 linearly dependent row vectors, 171 linearly independent column vectors, 171 linearly independent elements, 188 linearly independent row vectors, 171 INDEX 1483 linearly independent solutions of difference equation, 882 lines asymptotic, 393 condition for two lines in space to meet, 143 curvature, 393 curvilinear, 387 in space, theorem on arrangement, 136 parallel, 59, 96 perpendicular, 95 skew, 59 Liouville equation, 664 Liouville formula, 473, 518 for system of ODEs, 540 Liouville theorem, 403, 414 Lipschitz condition, 248, 310, 313, 454, 488, 543, 584 Lipschitz convergence criterion for Fourier series, 358 Lipschitz criterion of uniform convergence of Fourier series, 361 Lobachevsky convergence criterion for series, 341 Lobachevsky equation, 1444 local maximum, 257 point, 269 local minimum, 257 point, 269 local theorem of de Moivre–Laplace, 1038 log-normal distribution, 1042 logarithm, xxxi, 9 common, 9 decadic, 9 natural, xxxi logarithmic branch point, 415 logarithmic derivative of gamma function, 949, 954 logarithmic function, 22 graphs, 22 properties, 23 logarithmic identity, 9 logarithmic integral, 941, 958 logarithmic kernel, 801 logarithmic residue, 416 logarithmic spiral, 86 logarithms, natural, 9, 22, 23, 238 logistic difference equation, 875 longitude, 119 Lorentz transformation, 718, 1278 lower bound, 235 lower half-plane, 79 lower limit of sequence, 240 lower price of game, 1025 lower triangular matrix, 168 Lyapunov central limit theorem, 1071 Lyapunov function, 549 Lyapunov stability, 546 Lyapunov stable, 546 Lyapunov theorem of asymptotic stability, 549 of stability, 549 M Macdonald function, 953 Maclaurin formula, 257 Maclaurin method, 164 Maclaurin power series, 352 Maclaurin series, 353 Maclaurin–Cauchy integral criterion of conver- gence of series, 339 main diagonal, 168 major axis of ellipse, 98 majorant series, 349 many-valued function, branch, 404 mapping, 226 conformal, 419 groups, 227 groups, isomorphic, 227 invertible, 226 linear-fractional, 422 onto set, 226 range, 226 schlicht, 402 marginal distribution function, 1064 marginal probability functions, 1059 Markov inequality, 1070 Markov process, 1075 Markov property, 1075 mass of body of variable density, 327 mathematical induction, 16 mathematical model of transportation problem, 1019 mathematical physics equations, nonlinear, 1301 mathematical programming, 1012 general problem, 1012 mathematical statistics, 1081 problems, 1081 Mathieu equation, 980, 1221 modified, 982, 1220 Mathieu functions, 980, 1221 modified, 982 matrices, 167 commuting, 169 compatible size, 169 congruent, 182 conjunctive, 183 difference, 169 direct sum, 175 elementary, 181 elementary transformations, 180 equal, 168 equivalent, 180 Kronecker product, 175 multiplication, 169 similar, 181 subtraction, 169 sum, 168 unitary transformations, 183 169 matrix, 167 additively inverse, 169 adjoint, 170 1484 INDEX matrix (continued) adjugate, 180 antihermitian, 168 antisymmetric, 168 block, 174 canonical form, 181 canonical Jordan form, 182 characteristic, 183 characteristic equation, 185 characteristic polynomial, 185 complex conjugate, 170 correlation, 1066 covariance, 1065 defect, 177, 178 determinant, of order n, 176 determinant, of order 2, 175 determinant, of order 3, 176 diagonal, 168 dimension, 167 eigenvalue, 183 elementary transformation, 199 finite, 167 generalized permutation, 168 Hermitian, 168 identity, 168 index, 182 infinite, 167 inverse, 171 inverse left, 171 inverse right, 171 involutive, 172 multiplication by scalar, 169 nilpotent, 172 nondegenerate, 171 nonsingular, 171 normal, 171 normal form, 181 null, 167 of bilinear form, 215 of linear operator, 208 of linear system of algebraic equations, basic, 197 of moments, 1065 of sesquilinear form, 214 of simple structure, 182 opposite, 169 orthogonal, 170 payoff, 1024 payoff, saddle point, 1025 rank, 177, 178 real, 171 rectangular, 177 self-adjoint, 168 signature, 182 skew-Hermitian, 168 skew-symmetric, 168 spectrum, 183 square, 168, 176 subdiagonal, 168 superdiagonal, 168 symmetric, 168 matrix (continued) trace, 171 transforming, 181 triangular, lower, 168 triangular, strictly lower, 168 triangular, strictly upper, 168 unitary, 171 upper triangular, 168 Vandermonde, 179 zero, 167 matrix decomposition, 201 matrix entry, 167 cofactor, 176 maximin principle, 1025 maximum conditional, 270 constrained, 270 local, 257 local weak, 993, 1001, 1002, 1006 of function, xxxi, 257 strong, 1006 strong local, 994 strong of functional, 991 weak of functional, 991 maximum likelihood estimator, 1092 maximum modulus principle, 403 maximum principle, Pontryagin, 1011 mean functional, 13 geometric, 13 harmonic, 13 sample, 1087 mean curvature, 394 mean value of function, 318, 325 mean value theorems, 295 for double integral, 318 for triple integral, 325 mean values, general approach, 13 mean-square convergence, 315 mean-square deviation, 1088 mean-square regression, 1063, 1066 median, 1045 of trapezoid, 54 of triangle, 45 Mehler–Fock transform, 449 Meijer inversion formula, 447 Meijer transform, 447 Meijer transform (K-transform), 449 Mellin transform, 441, 449, 826, 842, 844 inverse, 442 properties, 442 tables, 1187–1190 Menelaus theorem, 49 meromorphic function, 414 Mertens theorem, 343 method averaging, 503 backward substitution in Gaussian elimination, 200 bisection, 261 Bubnov–Galerkin, for integral equations, 850 Bubnov–Galerkin, for ODEs, 509 INDEX 1485 method (continued) Budan–Fourier, 166 characteristics, generalized classical, 582 Clarkson–Kruskal direct, 708–710 Cole–Kevorkian, 504 collocation, 509, 523, 847, 848, 860 collocation, convergence theorem, 524 collocation, for integral equations, 847, 848 collocation, for ODEs, 509, 523 D’Alembert, 533 D’Alembert, for nonhomogeneous higher-order linear systems, 533 differential constraints, 737 estimation, 1091 false position, 262 Fourier, 602 Galerkin, 508 Gaussian elimination, backward substitution, 200 integral transforms, 611 Jacobi, 217 Lagrange, 217 Lagrange–Charpit, 573 Laplace cascade, 600 Lavrentiev regularization, 827 least-squared error, 511 least squares, 509, 1092 Lindstedt–Poincar ´ e, 502 Maclaurin, 164 mathematical induction, 16 minimal cost, 1020 modified Newton–Kantorovich, 859, 868 moments, 509, 1091 Newton, 164 Newton–Kantorovich, 859, 868 Newton–Raphson, 262 of accelerated convergence in eigenvalue problems, 513 of approximating kernel by degenerate one, 844 of argument elimination, for functional equations, 928 of differentiation for integral equations, 857, 866 of differentiation for ODEs, 857 of differentiation in independent variables, for functional equations, 925 of differentiation in parameter, for functional equations, 922 of Euler polygonal lines, 471 of functional separation of variables, 697 of generalized separation of variables, 681 of integral transforms, 825, 856, 865 of integral transforms, for linear PDEs, 611 of integral transforms, for nonlinear integral equations, 856, 865 of integration by differentiation, 465 of inverse scattering problem, 755 of least-squared error, 511 of least squares, 509, 849, 1092 of least squares, for integral equations, 849 method (continued) of least squares, normal system, 850 of matched asymptotic expansions, 506 of model solutions, 818 of partitioning domain, 511 of potentials, 1021 of quadratures, 808, 823 of quadratures, for integral equations, 808, 823, 852, 861, 870 of regular expansion in small parameter, for ODEs, 470, 501 of replacing kernel by degenerate kernel, 844 of scaled parameters, 502 of separation of variables, 602 of separation of variables, for first-order PDEs, 571 of separation of variables, for linear PDEs, 602 of substituting particular values of independent arguments, for functional equations, 926 of successive approximations, 469, 511, 832, 858 of symmetry reductions, classical, 732 of symmetry reductions, direct, 708 of symmetry reductions, nonclassical, 732, 747 of Taylor series expansion in independent variable, for ODEs, 470 of two-scale expansions, 504 of undetermined coefficients, 17, 277 of vanishing viscosity, 564, 580 parametric for defining curve, 123 parametric for defining surface, 123 penalty function, 1027 perturbation, 499 Picard, 469 projection, 508 quadrature, 861 quadrature, for integral equations, 870 quadric, 148 regula falsi, 262 regularization, for integral equations, 827 Runge–Kutta, 472, 512 shooting, 512 similarity, 669 simplex, 1014 single-step, of second-order approximation, 471 splitting, 704 step, 884, 885, 889, 893, 919 Sturm, 166 successive approximation, 822, 833, 858, 867 summation of series, 344 Tikhonov regularization, 828 Titov–Galaktionov, 693 Van der Pol–Krylov–Bogolyubov, 503 methods numerical for ODEs, 511, 512 of defining function, 236 of integral transforms, 841 perturbation, 499–507 projection, 508 metric space, 196 metric tensor, 1193 metric tensor components, 1195 1486 INDEX Meusnier theorem, 392 midline, 44 minimal cost method, 1020 minimal polynomial, 186 minimal surface, 394, 770 minimality of variance of estimator, 1090 minimax principle, 1025 minimax property, 210 minimax theorem for antagonistic two-person zero-sum games, 1026 von Neumann, 1026 minimum conditional, 270 constrained, 270 local, 257 local, strong, 994 local, weak, 993, 1001, 1002, 1006 of function, xxxi, 257 strong, 1006 strong, of functional, 991 weak, of functional, 991 Minkowski inequality for finite sums, 14 for integrals, 296 for series, 343 for vectors, 121, 193 minor, 177 basic, 178 cofactor, 177 first kind, 177 of rectangular matrix, 177 of square matrix, 176 second kind, 177 minor axis of ellipse, 98 mixed boundary value problem, 480, 593 for hyperbolic equations, 627 for PDEs, 593 mixed strategy, 1025 game price, 1026 mode of continuous probability distribution, 1045 of discrete probability distribution, 1045 model of transportation problem, 1020 of transportation problem, closed, 1020 model equation of gas dynamics, 558, 1254 modified associated Legendre functions, 966 modified Bessel equation, 1216 modified Bessel functions, 953 first kind, xxxi, 953, 1216 second kind, xxxi, 953, 1216 modified Korteweg–de Vries equation, 759 modified Mathieu equation, 982, 1220 modified Mathieu functions, 982 modified Newton–Kantorovich method, 859, 868 modified reciprocal equation, 160 modulus complementary, 971 elliptic, 971 of complex number, xxxii, 400 modulus (continued) of real number, xxx, 8 of vector, xxx modulus set, 907 Mollweide formulas, 44 moment, 1043 absolute, 1044 central, 1043 central, of n-dimensional random variable, 1065 initial, of multivariate random variable, 1065 initial, of two-dimensional random variable, 1060 kth absolute, 1044 kth central, 1043 kth sample, 1087 kth sample, central, 1087 of multivariate random variable, 1065 of two-dimensional random variable, 1060 sample, 1087 sample, central, 1087 moment method, 509 moment-generating function, 1047 moments of inertia of body, 327 of flat plate, 324 momentum conservation law, 996 Monge–Amp ` ere equation, 654, 1326 homogeneous, 668, 1326 monotone function, 236 monotone sequence, 238 Morera theorem, 410 Morley theorem, 49 Morse potential, 1278 motion of mass point, 87 movable critical points, 495 movable singularities of solutions of ordinary differential equations, 494 moving trihedral, 382 of curve, 382 multidimensional stochastic process, 1072 multifactor analysis of variance, 1108 multilinear form, 215 skew-symmetric, 215 symmetric, 215 multimodal distribution, 1045 multiple, common least, 4 multiple correlation coefficient, 1066 multiple integral, 328 asymptotic formulas, 328 multiple integration, 289 multiplication index, 12 of asymptotic series, 364 of matrices, 169 of matrix by scalar, 169 of polynomials, 155 of real numbers, 6 of series, 343 multiplicative inverse of real number, 6 multiplicative separable solution, 571, 596, 678 . conservation, 996 of conservation of mass, 784 of cosines, 44 of cosines of angles, 73 of cosines of sides, 73 of large numbers, 1069, 1070 of sines, 44, 73 of tangents, 44 weak of large numbers,. 471 of functional separation of variables, 697 of generalized separation of variables, 681 of integral transforms, 825, 856, 865 of integral transforms, for linear PDEs, 611 of integral transforms,. kernel, 844 of scaled parameters, 502 of separation of variables, 602 of separation of variables, for first-order PDEs, 571 of separation of variables, for linear PDEs, 602 of substituting particular