Handbook of mathematics for engineers and scienteists part 220 pps

INDEX 1501 solution additive separable, 571, 596, 597, 678 asymptotic for ODEs, 477 asymptotically stable, 546 Boussinesq, 1294 Cardano, 158 Cauchy problem, fundamental, 616, 617 classical, 594 D’Alembert for wave equation, 600 Descartes–Euler, 160 discontinuous for quasilinear equations, 558 equilibrium, 874, 884 exponential self-similar, 675 feasible, 1012 Ferrari, 161 formal, 594 functional separable, 697 functional separable, special, 697 fundamental, 599, 615 fundamental for Cauchy problem, 616, 617 fundamental for elliptic equations, 642 fundamental for system of ODEs, 539 general for difference equation, 874, 884 general for first-order PDEs, 553 general for ODEs, 453, 472, 488 generalized, 563 generalized, for first-order PDEs, 563, 568 generalized, Hopf formula, 565 generalized self-similar, 677 generalized separable, 681, 682 generalized stable, 569 generalized traveling-wave, 698 generalized viscosity, 579, 580 in form of infinite series, 597 invariant, 669, 674, 716, 724 L ´ evy, 914 limiting self-similar, 675 multiplicative separable, 571, 596, 678 N-soliton, 764, 765 of eigenvalue problems, 604 of system of linear algebraic equations, 198 one-parameter, 724 one-soliton, 765 optimal, 1012 particular, 453, 472, 518, 594, 599, 604 self-similar, 669, 1440 self-similar, continuous, 781 self-similar, exponential, 675 self-similar, generalized, 677 self-similar, limiting, 675 separable, 596 separable, generalized, 681 separable, multiplicative, 571 singular for first-order ODEs, 455 special functional separable, 697 stable generalized, 569 traveling-wave, 554, 667, 775, 790, 796 trivial, 197 unstable, 546 viscosity, 564 weak, 563 solutions fundamental system, 477, 478, 522, 523 linearly independent of difference equation, 882 of homogeneous system, fundamental system, 198 Sommerfeld conditions, 1288, 1289 source, 536, 538 spiral, 537 space affine, 189, 190 Banach, 196 complete, 196 continuous curve, 123 continuous surface, 123 coordinate axes, 114 coordinate system, 114 curve, 120 Euclidean, 192, 193 Euclidean, complex, 195 Euclidean, real, 192 Hilbert, 196 linear, 187–189 linear, complex, 188 linear, infinite-dimensional, 189 linear, real, 188 metric, 196 normed, 195 of elementary events, 1031 probability, 1032 probability, discrete, 1033 representation, 230 sample, 1031 surface, 119, 120 unitary, 195 vector, 187 space curves, 379 spaces Euclidean isomorphic, 194 linear isomorphic, 189 unitary isomorphic, 196 span, linear, 190 special Cauchy criterion, 341 special form, Laplace integral, 291 special functional separable solution, 697 special functions, 939–990 special linear group, 228 special orthogonal group, 229 special Riccati equation, 1208 special right-hand side, 815 specific Euclidean space, 192 specificity test, 1095 spectra of square matrices, 183 spectral decomposition of Hermitian operator, 212 of normal operator, 212 spectral density, 1075 spectral function of stationary process, 1075 spectrum of matrix, 183 sphere, 68, 224 diameter, 69 unit, 220 1502 INDEX spherical biangles, 71 spherical cap, 69 spherical coordinates, xxxi, 118, 119, 1199 spherical excess, 72 spherical geometry, 70 spherical sector, 69 spherical segment, 69 base, 69 spherical triangle, 71, 72 rectangular, 72 vertices, 71 spherical zone, 69 spheroid, 144 oblate, 144 prolate, 144 spiral hyperbolic, 86, 368 logarithmic, 86 of Archimedes, 85 spiral point stable, 537 unstable, 537 spiral sink, 537 spiral source, 537 splitting in derivatives, 719 splitting method, 704 square, scalar, 120 square integrable function, 314 square matrix, 168 minor, 176 trace, 171 square system of linear algebraic equations, 197 stability asymptotic, 546 Lyapunov, 546 stability condition, 568 for generalized solution, 568 stability property, 1055 stability theory, 546 stable focus, 537 stable generalized solutions, 569 stable node, 535, 538 stable spiral point, 537 standard deviation, 1045 standard form of linear programming problem, 1013 stationary equation heat (diffusion), anisotropic, 1323 heat (diffusion), with nonlinear source, 1320 Khokhlov–Zabolotskaya, 1322 stationary point, 220, 269, 545 stationary process, spectral function, 1075 stationary stochastic process, 1074 in narrow sense, 1074 in wide sense, 1074 stationary value of function, 220 statistic, 1084 Cochran, 1104 complete, 1090 Romanovskii, 1101 sufficient, 1090 statistical estimation, 1088 statistical estimator, 1088, 1090 statistical hypothesis, 1094 simple, 1096 statistical series, 1082, 1083 statistical test, 1095 statistical transformations, 1082 statistically independent events, 1036 steady transonic gas flow, 1324 Steklov inequality, 297 step method, 884, 885, 889, 893, 919 Stewart theorem, 48 Stieltjes integral, 312 Stieltjes integral sum, 312 Stirling formula, 944 stochastic process differentiation, 1073 in narrow sense, 1074 in wide sense, 1074 multidimensional, 1072 stationary, 1074 vector, 1072 with continuous time, 1072 with discrete time, 1072 stochastic processes, 1071–1073 complex, 1073 Stokes theorem, 334 straight line direction coefficients, 131 general equation, 90 in polar coordinates, 92 in projections in space, 133 in space, 59 in space, general equation, 132 intercept-intercept equation, 91 normalized equation, 91 on plane, 89 slope, 89 slope-intercept equation, 89 strategy, 1024 mixed, 1025 optimal, 1024 stream function, 1297 strengthened Jacobi condition, 1000, 1004, 1007 strengthened Legendre condition, 999, 1004, 1007 stress function, Airy, 1297 strictly concave function, 245 strictly convex function, 245 strictly hyperbolic system, 781, 790 strictly lower triangular matrix, 168 strictly upper triangular matrix, 168 strong law of large numbers, 1070 strong local minimum, 994 strong maximum, 1006 of functional, 991 strong minimum, 1006 of functional, 991 strong neighborhood, 991 Struve function, 448, 449 Student t-distribution, 1056 Student distribution, 1056 INDEX 1503 Student ratio, 1090 Sturges formula, 1083 Sturm comparison theorem on zeros of solutions, 487 Sturm method, 166 Sturm–Liouville problem, 482, 621, 622, 625, 626, 633 subdiagonal matrix, 168 subgroup, 227 compact, 229 continuous, 228 discrete, 228 finite, 228 normal, 227 submodulus set, 907 subsequence, 238 subspace improper, 190 invariant, 205 invariant of representation, 230 linear, 190 null, 190 of linear spaces, 190 orthogonal complement, 194 proper, 190 subspaces direct sum, 189 intersection, 190 sum, 189 substitution canonical, 465 Euler, 280 trigonometric, 280 trigonometric versatile, 282 subtraction of asymptotic series, 364 of hyperbolic functions, 37 of inverse hyperbolic functions, 41 of inverse trigonometric functions, 33 of matrices, 169 of trigonometric functions, 28 successive approximation method for integral equations, 822, 833, 858, 867 for ODEs, 469, 511 sufficient conditions of extremum, 269 sufficient estimators, 1090 sufficient statistic, 1090 sum Cauchy–Riemann, 286 integral, 286, 317 of complex numbers, 399 of elements of linear space, 187 of linear operators, 204 of matrices, 168 of polynomials, 155 of series, 337 of subspaces, 190 of vectors, 113 partial of series, 337 Stieltjes integral, 312 summation, formulas for trigonometric series, 362 summation, index, 12 sums functional, finite, 1116 involving hyperbolic functions, 1116 involving trigonometric functions, 1117 numerical, finite, 1113–1116 superdiagonal matrix, 168 superposition principle, 519, 880, 883 for homogeneous system, 528, 539 for nonhomogeneous system, 541 support line, 1014 supremum, xxxi sure event, 1031 surface center, 143, 151 central, 143 conical, 67 continuous in space, 123 cylindrical, 65 first quadratic form, 390 in Cartesian coordinate system, 119 in space, 119, 120, 123 intrinsic geometry, 391 minimal, 394, 770 normal, 153 of constant curvature, 394 oriented, 333 parametrized, 387 ruled, 146 second quadratic form, 392 tangent line, 388 Sylvester criterion, 219 Sylvester inequality, 178 Sylvester theorem, 187 symbol, Levi-Civita, 179 symbols Christoffel, of first kind, 396 Christoffel, of second kind, 396 of order O and o, 243 symmetric bilinear form, 214 symmetric integral equation, 835 symmetric kernel, 825, 829 symmetric linear programming problem, 1013 symmetric matrix, 168 symmetric multilinear form, 215 symmetric operator, 207 nonnegative, 207 nonpositive, 207 symmetric polyhedral angles, 61 symmetric random variable, 1041 symmetries differential equations, 716 nonclassical, 733 nonlinear second-order equations, 719 symmetry, 422 symmetry principle, 426 symmetry reductions based on generalized separation of variables, 712 classical method, 732 direct method, 708 nonclassical method, 732, 747 1504 INDEX system algebraic equations, linear, 197 characteristic, 553 Clairaut, 1238 Ermakov generalized, 1240 first-order linear homogeneous ODEs, 528, 529 first-order ordinary differential equations, 1227 hyperbolic, 781, 790 incomplete of eigenfunctions, 836 characteristic values, 836 conservation laws, 780 eigenfunctions, 836 eigenfunctions, complete, 836 Fredholm integral equations of second kind, 854 general form, autonomous, 1238 linear algebraic equations, 197 linear algebraic equations, consistent, 197 linear algebraic equations, determined, 197 linear algebraic equations, homogeneous, 197 linear algebraic equations, inconsistent, 197 linear algebraic equations, nonhomogeneous, 197 linear algebraic equations, square, 197 linear algebraic equations, underdetermined, 197 linear constant-coefficient ODE, 528 linear constant-coefficient ODE, first-order, 1239 linear equations, elementary transformation, 199 ODEs, autonomous, 545 ODEs, characteristic equation, 529 ODEs, first integral, 544 ODEs, fundamental solutions, 539 ODEs, Liouville formula, 540 ODEs, Wronskian determinant, 539 random variables, 1057 second-order ODEs, 1232 solved for derivative (for ODEs), 542 strictly hyperbolic, 781, 790 two algebraic equations in two unknowns, 198 two constant-coefficient first-order linear ODEs, 1229 two constant-coefficient second-order linear ODEs, 1232 systems linear ODEs, 1229–1237 linear PDEs, 1341, 1342 nonlinear ODEs, 1239–1246 nonlinear first-order PDEs, 1336–1341 nonlinear PDEs, of general form, 1374–1382 nonlinear second-order PDEs, 1343–1373 T t-distribution, 1056 tangency, point, 56 tangent, xxxi, 24, 26, 56, 368, 380 positive sense, 369 slope, 368 tangent line, 251 to surface, 388 tangent plane, 268, 381, 389 to quadric, 152 to surface, 152 tangent to second-order curve, 111 tangential curvature vector, 392 Taylor formula, 257, 266 remainder, 257 Taylor power series, 352, 410, 411 telegraph equation, 1282 terminal cost function, Bolza, 1001 terminal value problem, 580, 583, 584 termination point, 373 terms, secular, 502 test, 1095 Bartlett, 1103 Cauchy, 303 Cochran, 1104 Fisher, 1104 Kolmogorov, 1099 likelihood ratio, 1097 Pearson, 1100 Romanovskii, 1101 Smirnov, 1099 statistical, 1095 Wald, 1097 test power, 1096 test size, 1095 test specificity, 1095 theorem Abel, 343, 411 Abel, on convergence of power series, 351 Abel–Ruffini, 162 absolute value, 296 absolute value, for double integral, 318 absolute value, for triple integral, 325 addition, 1055 algebra, fundamental, 162 Bernoulli, 1070 binomial, 10 Bolzano–Cauchy, 237 boundary correspondence, 419 Brauer, 877 Casoratti, 902 Cauchy, 254 Cauchy, multiply connected domain, 409 Cauchy, residue, 416 Cauchy, series, 343 Cauchy, simply connected domain, 409 Cayley–Hamilton, 186, 212 central limit, 1071 Ceva, 48 Chebyshev, 1070 Chetaev, of instability, 550 continuity, for characteristic functions, 1067 convergence of collocation method, 524 convolution, for Fourier transform, 444 convolution, for Laplace transform, 437 curl, 334 de Moivre–Laplace, integral, 1038 INDEX 1505 theorem (continued) de Moivre–Laplace, limit, 1048 de Moivre–Laplace, local, 1038 decomposition, 208, 1055 density multiplication, 1062 Descartes, 165 Dirichlet, 295, 358 Dirichlet, representation of function by Fourier series, 357 divergence, 334 double integral, mean value, 318 duality, second, 1018 estimation for double integral, 318 estimation for triple integral, 325 Euler, 61, 393 Euler–Lagrange, 1009 existence and uniqueness, Cauchy problem for ODEs, 488, 524 existence and uniqueness, for first-order ODEs, 454 existence and uniqueness, for first-order PDEs, 557, 577 existence and uniqueness, for systems of ODEs, 543 Feller, 1077 Feuerbach, 49 first duality, 1018 first mean value for integration, 295 Fredholm, 834 fundamental of algebra, 162 Gakhov first, 432 Gakhov second, 433 Gauss, 335 Green, 335 half-side, 73 Hilbert–Schmidt, 837 integral of de Moivre–Laplace, 1038 Jacobi, 180 Kolmogorov, 1070 Kronecker–Capelli, 198 Lagrange, 254 Laplace, 177 Laplace transform, convolution, 437 Laplace transform, limit, 440 Legendre, 72 limit, for Bernoulli process, 1038 limit, for Laplace transform, 440 Lindeberg central limit, 1071 Liouville, 403, 414 local of de Moivre–Laplace, 1038 Lyapunov, central limit, 1071 Lyapunov, of asymptotic stability, 549 Lyapunov, of stability, 549 mean value, 295 mean value for double integral, 318 mean value for triple integral, 325 Menelaus, 49 Mertens, series, 343 Meusnier, 392 minimax, for antagonistic two-person zero-sum games, 1026 theorem (continued) minimax, von Neumann, 1026 Morera, 410 Morley, 49 Neyman–Fisher, 1090 of arrangement of two lines in space, 136 of boundary correspondence, 419 of projections, 44 on basic minor, 178 Ostrogradsky–Gauss, 334 Peano existence for system of ODEs, 543 Poisson, 428, 1070 polynomial remainder, 156 probability multiplication, 1035 projections, 44 Ptolemy, 52 Pythagorean, 50, 193 Riemann, 419 Riemann–Schwarz, 426 Rolle, 254 Rouch ´ e, 413 Schwarz–Christoffel, 427 second duality, 1018 second mean value for integration, 295 Sokhotskii, 431 Stewart, 48 Stokes, 334 Sturm comparison, 487 Sylvester, 186, 187 triple integral, mean value, 325 uniqueness, 413 Vi ` ete, 158, 159, 163 von Neumann minimax, 1026 theorems Fredholm, 834 of differentiable functions, 254 of instability, 549 of stability, 549 theoretical distribution function, 1082 theory game, 1024 stability, 546 theta functions, 978 third boundary value problem for elliptic equations, 634 for hyperbolic equations, 627 for ODEs, 481 for PDEs, 593 third Painlev ´ e transcendent, 497 third-order nonlinear PDEs, 1327 three-dimensional Helmholtz equation, 600 three-dimensional Laplace equation, 643, 644 three-dimensional Poisson equation, 599 three-dimensional wave equation, 618 throat ellipse, 145 Tikhonov regularization method, 828 Titov–Galaktionov method, 693 toroidal coordinates, 1205 torsion, 385 of space curves, 385 torsion radius, 385 torus, 68, 70 1506 INDEX total derivative, 265 total increments of function, 264 total probability formula, 1036 total variation of function, 247 trace of linear operator, 209 trace of matrix, 171 trajectory, orthogonal, 379 transcendent, Painlev ´ e, 495 transcendental branch point, 415 transform, 435 Bochner, 449 Euler, of first kind, 449 Euler, of second kind, 450 Fourier, 443, 449, 614, 843 Fourier, asymmetric form, 444 Fourier, n-dimensional, 445 Fourier, standard form, 443 Fourier cosine, 445, 449 Fourier inverse, 443, 614 Fourier sine, 445, 446, 449 Gauss, 449 Hankel, 446, 449, 1398 Hardy, 449 Hartley, 449 Hilbert, 449, 1396, 1398 integral, 435 inverse, 435 Kontorovich–Lebedev, 448, 449 Laplace, 436, 449, 611, 804, 813, 843, 856 Laplace, inverse, 437 Laplace, two-sided, 449 Meijer (K-transform), 447, 449 Mellin, 441, 449, 826, 842, 844 Mellin, inverse, 442 Weber, 449 Weierstrass, 449 transformation Abel, 345 Appell, 1266 auto-B ¨ acklund, 663 B ¨ acklund, 497, 663–665 Cole–Hopf, 751 congruence, 182 congruent, 182 conjunctive, 183 contact, 660 equivalence, 180 Euler, 661 Fisher, 1105 Galileo, 718 Gauss, 974 graphs of functions, 259 hodograph, 455, 657, 658 Hopf–Cole, 666, 1308 in plane, 717 invertible, 267 Kummer, 345, 957 Kummer–Liouville, 475 Landen, 974, 975 Landen, ascending, 975 Landen, descending, 974 transformation (continued) Legendre, 469, 660, 663 linear, 656 Lorentz, 718, 1278 of equation of quadric to canonical form, 152 of matrices, orthogonal, 182 one-parameter, 716 orthogonal, 182 orthogonal, improper, 229 point, 655 proper orthogonal, 229 scaling, 656, 676, 923 similarity, 181 simplifying boundary conditions, 649 simplifying initial conditions, 649 translation, 656, 668 unitary, 183 von Mises, 667 transformation group, 228 transforming matrix, 181 transition point, 477 transition probability, 1075 translation parallel, 220 transformation, 656, 668 transportation, unbalanced, 1020 transportation problem, 1019, 1020 closed model, 1020 mathematical model, 1019 open model, 1020 transpose, 170 of matrix, 170 of operator, 207 transposed equation, 825, 829, 834 transposition, 178 transversality condition, 577, 1001 transverse vibration of elastic rods, 1294 trapezoid, 54 altitude, 54 base, 54 isosceles, 54 leg, 54 median, 54 trapezoidal rule, 315, 808, 810, 824 computation of integrals, 315 traveling-wave equation, 1439 traveling-wave solution, 554, 667, 775, 790, 796 generalized, 697 trefoil, 373 trials, independent, 1037 triangle, 43 altitude, 46 area, 48 center of gravity, 45 circumcircle, 46 equilateral, 43, 51 excircle, 47 incircle, 46, 47 isosceles, 43, 50 median, 45 obtuse, 43 orthocenter, 46 INDEX 1507 triangle (continued) plane, 43 right, 43, 50 right-angled, 43 scalene, 43 spherical, 71, 72 triangle inequality, 14, 44, 73, 193, 314 triangle rule, 113, 176 triangles Euler, 71 plane, 43 triangular decomposition, 180 triangular pyramid, 63 triangular representation, 201 triaxial ellipsoid, 144 Tricomi equation, 1403 trigonometric circle, 24 trigonometric form of complex number, 400 trigonometric functions, 24 differentiation formulas, 29 expansion in power series, 30 integration formulas, 30 of half argument, 29 of multiple arguments, 29 trigonometric series, 1122–1128 trigonometric substitutions, 280 trihedral moving, 382 natural, 382 right-handed, 121 trihedral angle, 60 edge, 60 face, 60 vertex, 60 triple cross product of vectors, 122 triple integral, 324 change of variables, 326 estimation theorem, 325 geometric applications, 327 mean value theorem, 325 physical applications, 327 properties, 324 triple point, 371 trivial representation of group, 230 trivial solution, 197 truncated cylinder, 66 truncated expansion, 749 truncated prism, 62 tube, cylindrical, 67 two-body problem, 1260 two-dimensional Helmholtz equation, 599 two-dimensional Klein–Gordon equation, 618 two-dimensional Poisson equation, 599 two-dimensional random variable initial moment, 1060 moment, 1060 two-dimensional random vector, distribution function, 1057 two-dimensional wave equation, 618 two-person game, 1024 two-sheeted hyperboloid, imaginary semiaxis, 144 two-sided Laplace transform, 449 two-term asymptotic expansions, 478 types of linear PDEs, 585 types of matrices, 167 U ultrahyperbolic equation, 590 ultraspherical polynomials, 987 umbilical point, 393 unbalanced transportation, 1020 unbiased estimator, 1089 unbounded set, 263 uncomputable integrals, 276 uncorrelated random variables, 1061 underdetermined system of linear algebraic equations, 197 undetermined coefficients method, 17, 277 uniform convergence of functions, 249 Weierstrass criterion, 349 uniform distribution, 1051 uniformly convergent functional series, 349 uniformly convergent series, 349 unimodal distribution, 1045 union of events, 1031 uniqueness and existence theorems for first-order ODEs, 454 uniqueness theorem, 171, 413 unit, imaginary, 399 unit normal vector, 389 unit sphere, 220 unit vector, 113 unitarity condition, 206 unitary group, 229 unitary matrix, 171 unitary operator, 206 unitary space, 195 isomorphic, 196 normed, 195 orthogonal elements, 195 unitary transformation, 183 of matrices, 183 universal invariant, 731 unknown quantities, 199, 201 unstable focus, 536, 537 unstable solution, 546 unstable spiral point, 537 upper bound, 235 upper half-plane, 79 upper limit of sequence, 240 upper price of game, 1025 upper triangular matrix, 168 Urysohn equation with degenerate kernel, 863, 864 Urysohn integral equation, 856 V value Cauchy principal, for singular integral, 311 characteristic, 829, 834 . 286, 317 of complex numbers, 399 of elements of linear space, 187 of linear operators, 204 of matrices, 168 of polynomials, 155 of series, 337 of subspaces, 190 of vectors, 113 partial of series,. 1009 existence and uniqueness, Cauchy problem for ODEs, 488, 524 existence and uniqueness, for first-order ODEs, 454 existence and uniqueness, for first-order PDEs, 557, 577 existence and uniqueness, for. 1070 Chetaev, of instability, 550 continuity, for characteristic functions, 1067 convergence of collocation method, 524 convolution, for Fourier transform, 444 convolution, for Laplace transform, 437 curl,