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A. D. Polyanin and A. V. Manzhirov Handbook of Mathematics for Engineers and Scientists Publisher: Chapman & Hall/CRC Press Publication Date: 27 November 2006 Number of Pages: 1544 Summary Features Preface Contents Index Index A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z A Abel convergence criterion -- functional series, 349 -- power series, 341 Abel differential equation -- first kind, 462 -- first kind, canonical form, 463 -- second kind, 464 -- second kind, canonical form, 465 Abel differential equations, 462--465 Abel functional equation, 914 Abel identity, 902 Abel integral equation, 1385 -- generalized, 1386 -- second kind, 1392 -- second kind, generalized, 1393 Abel theorem, 343, 411 -- on convergence of power series, 351 Abel transformation, 345 Abel--Dini convergence criterion for series, 341 Abel--Ruffini theorem, 162 abelian group, 226 abscissa axis, 78 abscissa of point, 79, 115 absence of aftereffect, 1075 absolute convergence, 342 absolute convergence of infinite products, 348 absolute extremum, 991 absolute moment, 1044 absolute value of real number, xxx, 8 absolute value theorem, 296 -- double integral, 318 -- triple integral, 325 absolutely convergent functional series, 348 absolutely convergent infinite product, 348 absolutely convergent series, 342 acceleration of convergence of series, 345 acceleration vector, 381 acceptance region, 1095 acute triangle, 43 addition -- asymptotic series, 364 -- hyperbolic functions, 37 -- inverse hyperbolic functions, 41 -- inverse trigonometric functions, 33 -- matrices, 168 -- real numbers, 6 -- trigonometric functions, 28 addition theorem, 1055 additive inverse of real number, 6 additive separable solution, 571, 596, 597, 678 additively inverse matrix, 169 additivity of conditional expectation, 1063 adjoint matrix, 170 adjoint operator, 206 adjugate matrix, 180 adjusted sample variance, 1088 admissible controlled process, 1010 admissible element, 1009 admissible extremal, 994, 998 admissible function, 993, 1002, 1006 affine coordinate system, 190 affine space, 189 -- axioms, 189 -- dimension, 190 -- point, 189 Airy equation, 955 Airy functions, 477, 955 -- first kind, 955 -- second kind, 955 Airy stress function, 1295 algebra, 155 algebra of sets, 1032 algebraic branch point, 415 algebraic curve, 87 -- on plane, 87 algebraic equation -- arbitrary degree, 161 -- binomial, 161 -- bounds for roots, 163 -- complete cubic, 159 -- cubic, 158 -- discriminant, 163 -- fourth-degree, 159 -- incomplete cubic, 158 -- linear, 157 -- nth-degree, 162 -- quadratic, 158 -- reciprocal, 160, 161 -- reciprocal modified, 160 -- roots and coefficients, 163 algebraic equations, 157--167 algebraic multiplicity, 209 -- eigenvalue, 184 alternative Fourier transform, 444 alternative Fredholm, 834 alternative hypothesis, 1094 altitude -- pyramid, 63 -- trapezoid, 54 -- triangle, 46 amplitude, 973 analogs Neper, 73 analysis -- sequential, 1097 -- variance, 1107 analytic continuation, 418 analytic function, 413 analytic geometry, 77 angle -- between coordinate axes and segments, 82 -- between elements, 193 -- between lines in space, 135 -- between planes, 137 -- between straight line and plane, 139 -- between two straight lines, 94 -- between vectors, 120 -- central, 55, 56 -- dihedral, 60 -- edge, 60 -- external, 43 -- face, 60 -- inscribed, 56 -- interior, 43 -- interior of polygon, 51 -- polar, 80 -- polyhedral, 60, 61 -- solid, 61 -- trihedral, 60 angle bisector of triangle, 45, 46 angle trisectors, 49 angles -- in space, 59 -- symmetric polyhedral, 61 angular point, 252 annihilating polynomial, 186 annulus, 58 antagonistic game, 1024 antagonistic two-person zero-sum game, 1024 -- minimax theorem, 1026 antiderivative, 273 antihermitian matrix, 168 antiperiodic functions, 885 antisymmetric matrix, 168 apex, 67 apex of pyramid, 63 apothem of regular pyramid, 64 Appell transformation, 1268 applicable surfaces, 395 applicate of point, 115 approximation, best mean-square, 1063, 1066 approximation function, 508 arbitrary curvilinear coordinate systems, 1195 arbitrary series, 341 arc -- geodesic, 70 -- length, 390 -- length differential, 390 -- regular, 379 arccosine, xxx, 32 arccotangent, xxx, 32 arcsine, xxx, 31 arctangent, xxx, 32 area -- polygon, 83 -- triangle, 83 argument, 236 -- complex number, 400 -- increment, 992 -- variation, 992 arithmetic, 3 arithmetic mean, 13 -- of functions, 297 arithmetic progression, 11 arithmetic root, 8 arithmetic sequence, 11 arithmetic series, 11 arm of hyperbola, 101 arrangement, 1033 -- with repetitions, 1034 artificial basic variables, 1016 ascending Landen transformation, 975 associated Legendre functions, 964, 965 -- first kind, 965 -- second kind, 965 associated linear space of affine space, 189 associated vector, 213 associative composition law, 225 asymmetric form of Fourier cosine transform, 445 asymmetric form of Fourier sine transform, 446 asymmetric form of Fourier transform, 444 asymmetry coefficient, 1045 asymptote, 241, 374 -- oblique, 242 -- vertical, 242 asymptotes, 241, 374 -- of hyperbola, 101 asymptotic cone, 144 asymptotic expansion, 363 asymptotic formulas -- calculation of integrals, 290 -- improper integrals with parameter, 307 -- Laplace integrals of general form, 292 -- Laplace integrals of special form, 291 -- power Laplace integral, 294 asymptotic lines, 393 asymptotic point, 373 asymptotic properties of Fourier coefficients, 359 asymptotic series, 363 -- addition, 364 -- division, 364 -- integration, 365 -- multiplication, 364 -- Poincare type, 363 -- subtraction, 364 asymptotic stability, 546, 549 asymptotic zero, 364 asymptotically stable solution, 546 asymptotically unbiased estimator, 1089 attractive fixed point, 908 augmented matrix, 198 auto-Backlund transformations, 663 autocorrelation function, 1072 automorphic function, 912 automorphism of group, 227 autonomous equation, 525, 1207, 1223, 1225 autonomous system -- equilibrium point, 545 -- general form, 1240 -- ODEs, 545 averaging method, 503 axioms of affine space, 189 axis, 77 -- coordinate of point, 77 -- coordinate system, 77 -- imaginary, 401 -- imaginary of hyperbola, 101 -- iteration, 876 -- ordinate, 78 -- polar, 80 -- principal, 152 -- principal of hyperbola, 101 -- principal of second-order curve, 109 -- real, 235 -- semiminor of ellipse, 98 axisymmetric steady hydrodynamic boundary layer, 712 B Babbage equation, 917, 1416, 1423 Backlund transformation, 497, 663--665 backward Kolmogorov equation, 1076 backward substitution in Gaussian elimination method, 200 balanced transportation problem, 1020 ball, 68 Banach problem, 1038 Banach space, 196 Bartlett test, 1103 base -- cylinder, 65 -- isosceles triangle, 50 -- natural (Napierian) logarithms, 9, 22, 23 -- pyramid, 63 -- spherical segment, 69 -- trapezoid, 54 base face of prism, 61 basic columns, 178 basic equation of information theory, 1440 basic inequalities for mean values, 13 basic integrals, 274 basic matrix of linear system of algebraic equations, 197 basic minor, 177, 178 basic properties -- equalities, 5 -- inequalities, 5 basic rows, 178 basic variable, 1015 -- artificial, 1016 basis -- canonical, 217 -- linear space, 189 -- orthonormal, 194 Bayes formula, 1036, 1063 Bellman equations, 1028 Bellman optimality principle, 1028 bending, 395 Bernoulli equation, 458, 1207 Bernoulli formula, 1037 Bernoulli numbers, 938 Bernoulli polynomials, 891, 988 Bernoulli process, 1037 Bernoulli scheme, 1037 Bernoulli theorem, 1070 Bertrand convergence criterion for series, 340 Bessel equation, 1216 -- modified, 1216 Bessel formula, 950 Bessel function, 947, 1216 -- first kind, xxxi, 947, 1216 -- second kind, xxxii, 947, 1216 -- third kind, 952 best mean-square approximation, 1063, 1066 beta function, 945 biangles, spherical, 71 biased estimator, 1089 bicylindrical coordinates, 1204 biharmonic equation, 1297 bilinear form, 214, 992 -- degenerate, 215 -- Euclidean space, 219 -- finite-dimensional space, 215 -- matrix, 215 -- nondegenerate, 215 -- polar, 216 -- rank, 215 -- skew-symmetric, 214 -- symmetric, 214 bilinear functional, 992 bilinear functional equations, 930 bilinear series of iterated kernels, 838 bimodal distribution, 1045 binomial algebraic equation, 161 binomial coefficients, xxx, 10, 937 binomial distribution, 1047 binomial theorem, 10 binormal, 382 bipolar coordinates, 1204 biquadratic equation, 159 birth--death processes, 1077 bisection method, 261 bisector, perpendicular, 46 bivariate random variable, 1060 -- central moment, 1060 -- expectation, 1060 block matrix, 174 blow-up regime, 1303 Bochner transform, 449 body -- coordinates of center of mass, 327 -- moments of inertia, 327 Bolza functional, 1001 Bolza Lagrangian, 1001 Bolza problem, 1001 Bolza terminal cost function, 1001 Bolzano--Cauchy theorem, 237 Born--Infeld equation, 659 bound -- lower, 235 -- upper, 235 boundary -- confidence, 1081 -- domain, 401 -- positive sense, 402 boundary conditions, 484, 591, 592 -- first kind, 484 -- homogeneous, 481, 591, 649 -- mixed, 486 -- nonhomogeneous, 481, 649 -- second kind, 485 -- third kind, 486 boundary correspondence principle, 420 boundary correspondence theorem, 419 boundary function, 419 boundary point of domain, 401 boundary value problem, 427 -- arbitrary cylindrical domain, 642 -- difference equations, 879 -- elliptic equations, 631 -- first, 480, 593, 627 -- Hilbert--Privalov, 432 -- hyperbolic equations, 623 -- mixed, 480, 593, 627 -- nonhomogeneous, 620, 624 -- ordinary differential equations, 480 -- parabolic equations, 645 -- rectangular domains, 640 -- second, 480, 593, 627 -- third, 480, 593, 627 -- with many space variables, 634 bounded function, 236 bounded operator, 204 bounded sequence, 237 bounded set, 263 bounded variation, 247, 313 Boussinesq equation, 709, 1332 Boussinesq solution, 1294 bracket, Jacobi--Mayer, 574 branch of many-valued function, 404 branch point, 414 -- function, 404 -- infinite order, 415 -- logarithmic, 415 -- order, 414 Brauer theorem, 877 break point, 373 broken extremal, 997, 999 Bubnov--Galerkin method -- for integral equations, 850 -- for ODEs, 509 Budan--Fourier method, 166 Bugaev convergence criterion for series, 341 Bunyakovsky inequality, 296 Burgers equation, 664, 666, 751, 1307 -- radial symmetric case, 1307 Burman--Lagrange formula, 356 C calculus of variations, 991 -- simplest problem, 993 canonical basis, 217 canonical coefficients, 217 canonical coordinates -- ellipse, 98 -- hyperbola, 101 -- parabola, 104 canonical equation -- circle, 97 -- ellipse, 98 -- hyperbola, 101 -- noncentral hypersurface, 224 -- parabola, 104 -- straight line in space, 132 -- straight line passing through given point on plane, 90 -- straight line passing through two given points on plane, 90 canonical form -- Abel differential equation, of first kind, 463 -- Abel differential equation, of second kind, 465 -- central surfaces, 143 -- elliptic equation, 588 -- gas dynamics systems, 786 -- hyperbolic equation, 587 -- Jordan, of matrix, 182 -- linear operators, 213 -- matrix, 181 -- parabolic equation, 586 -- quadric, 148 -- Riccati differential equation, 460 -- second-order curve, 107 canonical representation of quadratic form, 217 canonical second-order curves, 107 canonical substitutions, 465 Cantor equation -- first, 1442 -- second, 1449 cap, spherical, 69 Cardano solution, 158 Carleman equation, 1399 Cartesian coordinate system, 78, 114 -- in plane, 78 -- in space, 114 -- oblique, 79 Casoratti determinant, 901 Casoratti theorem, 902 Cauchy convergence criterion for series, 339 Cauchy criterion, 337 Cauchy distribution, 1054 Cauchy equation, 1438 -- exponential, 1443 -- logarithmic, 1439 -- power, 1444 Cauchy formula -- for multiple integration, 289 -- for ODE, 519 Cauchy inequality, 14, 296, 343 -- for functions of complex variables, 410 Cauchy integral formula, 409 Cauchy principal value for singular integral, 311, 430 Cauchy problem, 591, 592, 785, 1308 -- difference equations, 874, 889, 893 -- existence and uniqueness theorems for ODEs, 454, 488, 524, 577 -- first-order ODEs, 454 -- first-order PDEs, 556, 576 -- fundamental solution, 616, 617 -- generalized, 627 -- Hamilton--Jacobi equation, 577 -- hyperbolic equations, 617 -- parabolic equations, 615 -- second-order ODEs, 488 -- system of PDEs, 785 Cauchy series multiplication formula, 343 Cauchy tests, 303 Cauchy theorem, 254, 343 -- multiply connected domain, 409 -- residue, 416 -- simply connected domain, 409 Cauchy-type singular integral, 310, 409, 430 Cauchy--Bunyakovski inequality, 14, 296, 343 Cauchy--Hadamard formula, 411 Cauchy--Riemann conditions, 402 Cauchy--Riemann sum, 286 Cauchy--Schwarz inequality, 121, 193, 1043 Cauchy--Schwarz--Bunyakovski inequality, 14, 314 Cayley--Hamilton theorem, 186, 212 center -- curvature, 375 -- ellipse, 99 -- equilibrium point, for ODEs, 537 -- gravity of triangle, 45 -- hyperbola, 102 -- pencil, 92 -- regular polygon, 55 -- second-order curve, 109 -- second-order hypersurface, 222 -- surface, 143, 150 center of mass -- of body, 327 -- of flat plate, 324 centered rarefaction wave, 558 central angle, 55, 56 central cross-correlation function, 1073 central cylinder, 225 central hypersurface, 222 central limit theorem, 1071 central moment, 1043 -- n-dimensional random variable, 1065 -- bivariate random variable, 1060 central second-order curve, 109 central surface, 143 centroid, 45 Ceva theorem, 48 character of group element, 231 characteristic, 654 characteristic equation, 514, 654, 1046, 1066 -- difference equation, 881 -- linear operator, 211 -- linear PDEs, 585 -- matrix, 185 -- quadratic form, 150 -- second-order curves, 108 -- system of ODEs, 529 characteristic function, 1046, 1066 -- continuity theorem, 1067 characteristic index, 1221 characteristic matrix, 183 characteristic point, 377 characteristic polynomial -- linear operator, 211 -- matrix, 185 characteristic quadratic form, 150 -- of quadric, 150 characteristic system for first-order PDEs, 553, 555, 573, 576 characteristic value -- integral equation, 829, 834--836, 840 -- linear operator, 209 characteristic velocity, 790 Chebyshev formula, 808 Chebyshev functions of second kind, 985 Chebyshev inequalities, 14, 297, 1070 -- generalized, 15 Chebyshev nodes, 523 Chebyshev polynomials -- first kind, 983 -- second kind, 984 Chebyshev theorem, 1070 Chetaev theorem of instability, 550 chi-square distribution, 1054 chord, 56, 143 Christoffel symbols -- first kind, 396 -- second kind, 396 circle, 56, 97 -- canonical equation, 97 -- curvature, 375 -- diameter, 56 -- great of a sphere, 68, 70 -- nine-point, 49 -- osculating, 375 -- parametric equations, 98 -- small of a sphere, 71 circular point, 393 circular sector, 58 circulation, 331 circumcenter, 46 circumcircle, 46 -- of triangle, 46 circumscribed about circle, polygon, 51 cissoid of Diocles, 372 Clairaut equation, 467, 572, 1262 Clairaut system, 1240 Clarkson--Kruskal direct method, 708--710 classical definition of probability, 1033 classical integral functional, 993 classical method of symmetry reductions, 716 classical solution, 594 classification -- central second-order hypersurfaces, 223 -- noncentral second-order hypersurfaces, 224 -- points on surface, 394 -- quadrics, 148 -- second-order linear PDEs, 585 closed domain, 402 closed model of transportation problem, 1020 closed path integral, 331 coalition game, 1024 Cochran statistic, 1104 Cochran test, 1104 coefficient -- asymmetry, 1045 -- asymmetry, sample, 1088 -- correlation, multiple, 1066 -- excess, 1045 -- excess, sample, 1088 -- Pearson first skewness, 1045 -- reflection, 760 -- regression, 1063 -- variation, 1045 coefficients -- binomial, xxx, 10, 28 -- canonical for quadratic form, 217 -- Fourier, 358 -- Lame, 1196 -- system of linear equations, 197 -- undetermined, 847 cofactor -- of matrix entry, 176 -- of minor, 177 Cole--Hopf transformation, 751 Cole--Kevorkian scheme (method), 504 collinear vectors, 113 collocation method, 509, 523, 847, 848, 860 -- convergence theorem, 524 -- integral equations, 847, 860 collocation points, 848 column, pivot, 1016 column expansion of determinant, 176 column vector, 167 column vectors -- linear combination, 171 -- linearly dependent, 171 -- linearly independent, 171 columns, basic, 178 combinations, 1033 combinations, integrable, 545 -- with repetitions, 1034 common difference, 11 common logarithm, 9 common ratio, 11 commutative composition law, 225 commutative group, 226 commutator of operators, 755 commuting matrices, 169 compact subgroup, 229 compatibility conditions -- for PDEs, 738, 740, 742, 744 -- for systems of linear equations, 757 -- of initial and boundary conditions, 607, 609 compatible size of matrices, 169 complementary error function, xxx, 939, 958 complementary event, 1031 complementary modulus, 971 complete cubic equation, 159 complete elliptic integral, 969 -- first kind, 969 -- second kind, 969 complete equation of plane, see general equation of plane complete group of events, 1031 complete group of incompatible events, 1032 complete integral, 570 complete space, 196 complete statistic, 1090 complete system of eigenfunctions, 836 completely integrable, 773 completely reducible representation of group, 231 complex conjugate matrix, 170 complex Euclidean space, 195 complex linear space, 188 complex number, 399 -- argument, 400 -- conjugate, 399 -- exponential form, 405 -- geometric interpretation, 399 -- modulus, 400 -- root, 399 -- trigonometric form, 400 complex numbers -- difference, 399 -- equal, 399 -- product, 399 -- quotient, 399 -- sum, 399 complex plane, 401 -- domain, 401 -- extended, 402 complex separation of variables, 679 complex stochastic processes, 1073 components, metric tensor, 1195 composite function, 237, 265 composite hypothesis, 1094 composite number, 3 composition, 225 composition law, 225 compound percentage, 7 concave function, 245 condition -- boundary, 484, 591, 592 -- boundary, of first kind, 484 -- boundary, of second kind, 485 -- boundary, of third kind, 486 -- first-order necessary for extremum, 994 -- Holder, 310, 430 -- initial homogeneous, 591 -- invariance, 720, 732 -- invariant surface, 732 -- Jacobi, 998, 1000, 1004, 1007 -- Jacobi, strengthened, 1000, 1004, 1007 -- jump, 561 -- Lax, 792 -- Legendre, 998, 999, 1004, 1007 -- Legendre, strengthened, 999, 1004, 1007 -- line and plane to be parallel, 140 -- line and plane to be perpendicular, 140 -- line to be entirely contained in plane, 140 -- Lipschitz, 248, 310, 313, 543 -- necessary of extremum, 269 -- planes to be parallel, 138 -- planes to be perpendicular, 138 -- planes to coincide, 138 -- stability for generalized solution, 568 -- straight lines to be parallel on plane, 96 -- straight lines to be perpendicular on plane, 95 -- straight lines to coincide on plane, 96 -- three points to be collinear, 93 -- transversality, 577 -- two lines in space to meet, 143 -- two lines to be parallel in space, 135 -- two lines to be perpendicular in space, 136 -- two planes to be parallel, 138 -- two planes to be perpendicular, 138 -- unitarity, 206 -- vectors to be parallel, 122 -- vectors to be perpendicular, 122 -- Weierstrass, 998 conditional convergence, 342 conditional cumulative distribution function, 1062 conditional distribution, 1062 conditional expectation, 1063, 1066 conditional extremum of function, 271 conditional minimum, 270 conditional probability, 1035 conditionally convergent series, 342 conditions -- Cauchy--Riemann, 402 -- compatibility for PDEs, 738, 740, 742, 744 -- compatibility for systems of linear equations, 757 -- compatibility of initial and boundary conditions, 607, 609 -- d'Alembert--Euler, 402 -- Dirichlet, 357 -- evolutionary, 792 -- existence of extremum, 258 -- extremum of function of two variables, 269 -- radiation, 1290 -- Sommerfeld, 1290, 1291 -- sufficient of extremum, 269 -- transversality, 1001 -- Weierstrass--Erdmann, 997, 999 cone, 67, 68, 145 -- asymptotic, 144 -- frustum, 68 -- right circular, 68, 145 confidence boundary, 1081 confidence interval, 1093 confidence level, 1093 conformal mapping, 419 congruence transformation, 182 congruent matrices, 182 congruent polyhedral angles, 61 congruent transformation, 182 conic point, 391 -- of surface, 391 conical coordinates, 1202 conical surface, 67 conjugate complex number, 399 conjugate diameters, 151 -- of ellipse, 100 -- of hyperbola, 104 conjugate element of group, 231 conjugate matrix, 170 conjugate point, 998, 1000, 1004, 1007 conjunctive matrices, 183 conjunctive transformation, 183 conservation law, 564, 766, 996 -- at discontinuity, 561 -- for PDEs, 766 consistency condition for general linear system, 198 consistency of estimator, 1089 consistent system of linear algebraic equations, 197 constant -- Euler, xxx, 944, 949, 958 -- Holder, 310 -- separation, 678 constrained maximum, 270 constrained minimum, 270 constraint, 270 -- differential, 737, 738, 1009, 1010 -- differential, first-order, 739 -- differential, secod-order, 744 -- isoperimetric, 1002 contact transformation -- for ordinary differential equations, 468 -- for partial differential equations, 660 continuation, direct analytic, 418 continuity equation, 784 continuity theorem for characteristic functions, 1067 continuous bivariate random variable, 1059 continuous curve in space, 123 continuous distributions, 1051 continuous function, 243 -- of bounded variation, 249 continuous point group, 730 continuous probability distribution, mode, 1045 continuous random variable, 1040, 1064 -- expectation, 1042 continuous subgroups, 228 continuous surface in space, 123 contraction ratio, 403 control, 1010 controlled process, admissible, 1010 convergence -- almost sure, 1068 -- in distribution, 1069 -- in mean, 1068 -- in probability, 1068 -- mean-square, 315 -- nonuniform, 249 -- pointwise, 249 -- uniform, 249 -- weak, 1069 convergence criteria -- arbitrary series, 341 -- infinite products, 347 -- series with positive terms, 338 convergence criterion -- Abel, 341 -- Abel--Dini, 341 -- Bertrand, 340 -- Bugaev, 341 -- D'Alembert, 338 -- Dini, 341 -- Dirichlet, 342 -- Dirichlet--Jordan, 358, 361 -- Ermakov, 340 -- Gauss, 339 -- generalized Ermakov, 340 -- Kummer, 340 -- Lipschitz, 358 -- Lobachevsky, 341 -- Maclaurin--Cauchy, 339 -- Raabe, 339 -- Sapogov, 340 -- Weierstrass, 349 convergence domain of series, 348 convergence radius of power series, 350, 411 convergence theorem, collocation method, 524 convergence with probability 1, 1068 convergent infinite products, 346 convergent sequence, 237, 238 convergent series, 337, 338 convex function, 245 convex polygon, 51 convex polyhedron, 61 convex programming, 1027 convolution theorem, 437, 444 -- Fourier transform, 444 -- Laplace transform, 437 convolution-type equation, 825, 829 coordinate axes -- in space, 114 -- on plane, 78 coordinate functions, 847, 851 coordinate net, 387 coordinate of point on axis, 77 coordinate space, 188 coordinate system -- affine, 190 -- in space, 114 -- left rectangular Cartesian, 79 -- oblique Cartesian, 79 -- on axis, 77 -- on plane, 78 -- polar, 80 -- rectangular Cartesian, 78, 114 -- right rectangular Cartesian, 79 coordinate vectors, 388 coordinates -- bicylindrical, 326, 1204 -- bipolar, 1204 -- Cartesian, in plane, 79 -- Cartesian, in space, 115 -- center of mass of body, 327 -- center of mass of flat plate, 324 -- conical, 1202 -- current, 84 -- curvilinear, 387 -- cylindrical, xxxi, 118, 326, 1198 -- element, 188, 189 -- ellipse canonical, 98 -- elliptic, 1202 -- elliptic cylinder, 1202 -- first prolongation, 718 -- Gaussian, 387 -- general orthogonal curvilinear, 1196 -- hyperbola canonical, 101 -- line, 77 -- oblate ellipsoid of revolution, 1200 -- orthogonal, 1196 -- parabola canonical, 104 -- parabolic, 1203 -- parabolic cylinder, 1203 -- plane, 78 -- point, 79, 115 -- polar, 80, 1198 -- second prolongation, 718 -- spherical, xxxi, 119, 326, 1199 -- toroidal, 326, 1205 coplanar vectors, 113 correlated random variables, 1061 correlation -- function, 1072 -- matrix, 1066 -- random variables, 1061 -- sample, 1104 -- two random variables, 1061 correspondence, isometric, 395 cosecant, xxx, 24 coset, 227 -- left, 227 -- right, 227 cosine, 24, 25 cosine integral, 942 cost, reduced, 1015 cotangent, xxx, 24, 26 countable set, 3 counterexample, 17 covariance, 1060, 1061 -- matrix, 1065 -- of two random variables, 1061 Cramer rule, 199 Cramer--Rao efficient unbiased estimator, 1089 Cramer--Rao inequality, 1089 criterion -- Cauchy convergence for series, 339 -- Dini convergence for series, 341 -- Dirichlet convergence, 342, 350 -- Dirichlet--Jordan convergence for Fourier series, 358, 361 -- Ermakov convergence for series, 340 -- first comparison, 338 -- Gauss convergence for series, 339 -- Kummer convergence for series, 340 -- Leibnitz convergence for series, 341 -- Lipschitz convergence for series, 358, 361 -- Lobachevsky convergence for series, 341 -- Maclaurin--Cauchy convergence for series, 339 -- Routh--Hurwitz, 167 -- Sapogov convergence for series, 340 -- second convergence, 338 -- special Cauchy, 341 -- Sylvester, 219 -- Weierstrass convergence, 349 critical region, 1095 cross-correlation function, 1072 cross-covariance function, 1072 cross product of vectors, 121 cube, 62 cubic equation, 158 cumulative distribution function, 1039 -- conditional, 1062 cumulative frequency, 1084 cumulative relative frequency, 1084 curl, 331 -- of vector field, 331 -- theorem, 334 curvature, 376, 384 -- center, 375 -- circle, 375 -- extrinsic, 394 -- Gaussian, 394 -- geodesic, 392 -- lines, 393 -- mean, 394 -- normal, 392, 394 -- of curve on surface, 392 -- of plane curves, 376 -- of space curves, 384 -- radius, 376, 384 -- vector, 383 curve -- discriminant, 378 -- in space, 120, 123 -- intrinsic equations, 388 -- on plane, 84 -- on plane, algebraic, 87 -- parabolic type of second-order, 108 -- second-order, 107 curvilinear coordinates, 387 -- on surface, 387 curvilinear integral, 329 curvilinear lines, 387 cusp, first kind, 372 cusp, second kind, 372 cycle, 1022 cyclic functional equation, 920 cyclic groups, 229 cylinder, 65, 66 -- base, 65 -- central, 225 -- function, 947 -- paraboloidal, 225 -- right, 66 -- right circular, 66 -- round, 66 -- segment, 67 -- truncated, 66 cylindrical coordinates, xxxi, 118, 1198 cylindrical function, 947, 1214 cylindrical Korteweg--de Vries equation, 1328 cylindrical surface, 65 cylindrical tube, 67 D D'Alembert convergence criterion for series, 338 D'Alembert equation, 1443 D'Alembert formula, 1278 D'Alembert (Gauss) formulas, 73 D'Alembert method, 533 D'Alembert solution for wave equation, 600 D'Alembert--Euler conditions, 402 damped vibrations, 1213 Darboux equation, 458 Darboux vector, 386 de Moivre formula, 30, 400 de Morgan laws, 1032 death process, 1077 decadic logarithm, 9 decile, 1045 decomposition -- of matrices, 173, 201 -- theorems, 208, 1055 -- triangular, 180 decreasing function, 236 decreasing sequence, 238 defect of matrix, 178 definite integral, 286, 1147 -- geometric applications, 299 -- geometric meaning, 287 -- physical applications, 299 -- properties, 287 definite integrals -- involving exponential functions, 1150 -- involving hyperbolic functions, 1152 -- involving logarithmic functions, 1152 -- involving trigonometric functions, 1153 degenerate bilinear form, 215 degenerate hypergeometric equation, 1215 degenerate hypergeometric function, xxxii, 956 -- Wronskian, 958 degenerate kernel, 801, 802, 810, 811, 825, 829 degenerate matrix, 178 degenerate quadratic form, 216 density, spectral, 1075 density multiplication theorem, 1062 dependent variable, 236 derivation formulas, 396 derivative, 250 -- basic elementary functions, 252 -- directional, 267, 268 -- geometrical meaning, 251 -- higher-order, 255 -- of function, 402 -- of polynomial, 157 -- one-sided, 252 -- partial, 264 -- physical meaning, 251 -- total, 265 Descartes, Folium, 374 Descartes theorem, 165 Descartes--Euler solution, 160 descending Landen transformation, 974 determinant, 175 -- Casoratti, 901 -- column expansion, 176 -- Gram, 193 -- Jacobian, 321 -- linear operator, 209 -- order n, 176 -- order 2, 175 -- order 3, 176 -- product of matrices, 179 -- properties, 176 -- row expansion, 176 -- sum of matrices, 179 -- Vandermonde, 179 -- Wronskian, 473, 482, 518, 539 determined system of linear algebraic equations, 197 deviation -- of point from line, 93 -- of point from plane, 141 -- residual standard, 1063 -- root-mean-square, 1045 -- sample, 1088 -- standard, 1045 diagonal -- main, 168 -- secondary, 168 diagonal matrix, 168 diameter, 150 -- conjugate to chords, 108 -- conjugate to family of planes, 150 -- of circle, 56 -- of ellipse, 100 -- of hyperbola, 104 -- of parabola, 106 -- of second-order curve, 108 -- of set, 317 -- of sphere, 69 -- partition, 317 -- principal, of ellipse, 100 -- principal, of hyperbola, 104 -- principal, of parabola, 106 diameters -- conjugate, 151 -- conjugate, of ellipse, 100 -- conjugate, of hyperbola, 104 Dido problem, 1003 diet problem, 1012 difference -- common, 11 -- finite, 884 -- of complex numbers, 399 -- of events, 1031 -- of matrices, 169 -- of polynomials, 155 -- of vectors, 113, 188 difference equation, 889, 1409, 1422, 1425--1426 -- characteristic, 881 -- general solution, 874, 884 -- in integer argument, 873 -- linear, constant-coefficient, 873, 877 -- linear, higher-order, 1425 -- linear, second-order, 877, 1421 -- linear homogeneous, 881, 885, 889, 906, 904 -- linear homogeneous, first-order, 873 -- linear homogeneous, second-order, 877, 1421 -- linear nonhomogeneous, 883, 890--893 -- linear nonhomogeneous, constant-coefficient, first-order, 1411 -- linear nonhomogeneous, constant-coefficient, nth-order, 1427 -- linear nonhomogeneous, constant-coefficient, second-order, 1422 -- linear nonhomogeneous, first-order, 873 -- linear nonhomogeneous, nth-order, 1426 -- linear nonhomogeneous, second-order, 879, 880, 1421 -- linearly independent solutions, 882 -- logistic, 875 -- nonlinear, 884 -- nonlinear, first-order, 873 -- Riccati, 875, 918 difference kernel, 801, 804, 810, 813, 825, 829, 841 differentiable function, 251 differential, 250, 251, 265 -- higher-order, 255 differential constraint, 737, 1009, 1010 -- first-order, 739 -- second-order, 744 differential constraints method, 737 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 -- autonomous, 1207, 1223, 1225 -- axisymmetric steady hydrodynamic boundary layer, 712 -- Bernoulli, 458, 1207 -- Bessel, 1216 -- Bessel, modified, 1216 -- 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 -- characteristic, for system of ODEs, 529 -- Clairaut, 467, 572, 1262 -- Darboux, 458 -- degenerate hypergeometric, 1215 -- diffusion, nonlinear, with cubic source, 753 -- diffusion boundary layer, 1276 -- Duffing, 501 -- elliptic, 585, 588, 590, 1284 -- Emden--Fowler, 1223 -- Ermakov, 1222 -- Euler, 473, 520, 994, 998, 1216 -- evolution, second-order, 739 -- exact, first-order, 458 -- exact, second-order, 491 -- Fisher, 1299 -- Fitzhugh--Nagumo, 1302 -- free oscillations, 1213 -- gas dynamics, 558 -- Gaussian hypergeometric, 960, 1218 -- generalized homogeneous, 526 -- 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 -- homogeneous, 526, 591, 592, 810, 829, 834, 1207, 1224 -- homogeneous, generalized, 1210, 1224 -- hyperbolic, 585, 587, 590 -- hyperbolic, first canonical form, 587 -- hyperbolic, normal, 590 -- hyperbolic, second canonical form, 587 -- hypergeometric, 960, 1218 -- hypergeometric, degenerate, 1215 -- 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 -- Korteweg--de Vries, modified, 1328 -- Lagrange, 467 -- Laplace, ODE, 520 -- Laplace, three-dimensional, PDE, 643 -- Laplace, two-dimensional, PDE, 585, 633, 1284 -- Legendre, 1217--1219 -- Lienard, 1225 -- light rays, 1259 -- linear, constant-coefficient, second-order, 473, 1213 -- linear homogeneous, second order, 1212 -- Liouville, 664 -- Mathieu, 980, 1221 -- Mathieu modified, 982, 1220 -- minimal surfaces, 770 -- model of gas dynamics, 1254 -- Monge--Ampere, 1326 -- Monge--Ampere, homogeneous, 668, 1326 -- Monge--Ampere, nonhomogeneous, 1326 -- Newell--Whitehead, 733 -- ordinary, 453--550, 1207--1228 -- Painleve, 495 -- parabolic, 585, 586, 590 -- parabolic, in narrow sense, 590 -- partial, 553--798, 1247--1382 -- Pfaffian, 772, 773 -- Poisson, 1287 -- Poisson, three-dimensional, 599 -- Poisson, two-dimensional, 599 -- Riccati, 460 -- Riccati, general, 461, 462, 1209, 1210 -- Riccati, special, 1208 -- Schrodinger, linear, 1276 -- Schrodinger, nonlinear, 759, 1309 -- Schrodinger, of general form, 1311 -- Schrodinger, with cubic nonlinearity, 1309 -- Schrodinger, with power-law nonlinearity, 1310 -- separable, 456, 1207 -- separated, 456 -- sine-Gordon, 1314 -- sinh-Gordon, 1314 -- steady transonic gas flow, 1324 -- telegraph, 1284 -- telegraph, generalized, 626 -- thermal boundary layer, 1276 -- transonic gas flow, 1324 -- ultrahyperbolic, 590 -- vibration, with axial symmetry, 1282 -- vibration, with central symmetry, 1283 -- vibration of string, 1278 -- wave, 585, 1278 -- wave, nonlinear, 723, 1312--1317 -- wave, three-dimensional, 618 -- wave, two-dimensional, 618 -- Whittaker, 960 differential geometry, 367 differential operator -- linear, 591, 592 -- total, 740, 742 differential operators, of field theory, 272 differential substitution, 666 differentiation, 266 -- of implicit function, 266 -- of stochastic process, 1073 -- rules, 253 differentiation method for PDEs, 700 digamma function, 944, 949 dihedral angle, 60 -- edge, 60 -- face, 60 dilatation ratio, 403 dimension -- of affine space, 190 -- of linear space, 189 -- of matrix, 167 -- of representation, 230 Dini convergence criterion for series, 341 Dini formula, 429 Dini integral, 1287 Diocles cissoid, 372 Dirac delta function, 599 direct analytic continuation, 418 direct method of symmetry reductions, 708 direct scattering problem, 760 direct sum -- of subspaces, 191 -- of two matrices, 175 directed segment, 77 direction, principal of second-order curve, 109 direction coefficients of straight line, 131 direction cosine of segment, 116 directional derivative, 267, 268 directrix, 65, 67 -- of ellipse, 99 -- of hyperbola, 102 -- of parabola, 105 Dirichlet conditions, 357 Dirichlet convergence criteria, 341, 342 Dirichlet criterion of uniform convergence of functional series, 350 Dirichlet problem, 427, 428 -- for elliptic equations, 593, 1330 Dirichlet theorem, 295, 358 -- representation of function by Fourier series, 357 Dirichlet--Jordan convergence criterion for Fourier series, 358, 361 Dirichlet--Mehler integral, 963 discontinuity, conservation law, 561 discontinuous solutions of quasilinear equations, 558 discrete bivariate random variable, 1058 discrete probability distribution, 1045 discrete probability space, 1033 discrete random variable, 1039, 1064 -- expectation, 1042, 1063 discrete subgroups, 228 discriminant -- large, 108 -- of algebraic equation, 163 -- small, 108 discriminant curve, 378 disk, 56, 97 dispersion equation, 532, 596 distance -- between elements, 193, 195 -- between lines in space, 142 -- between parallel lines, 96 -- between points on axis, 78 -- between points on plane, 81 -- between two parallel planes, 142 -- from point to line in space, 142 -- from point to plane, 141 -- from point to straight line, 93 -- shortest, 70 distribution -- bimodal, 1045 -- binomial, 1037, 1047 -- Cauchy, 1054 -- chi-square, 1054 -- conditional, 1062 -- continuous, 1051 -- convergence, 1069 -- exponential, 1052 -- Fisher--Snedecor, 1086 -- geometric, 1048 -- hypergeometric, 1049 -- Kolmogorov, 1086 -- log-normal, 1042 -- multimodal, 1045 -- negative binomial, 1050 -- normal, 1053 -- Poisson, 1050 -- Student, 1056 -- uniform, 1051 -- unimodal, 1045 distribution function -- empirical, 1084 -- joint, 1064 -- marginal, 1064 -- multivariate random variable, 1064 -- two-dimensional random vector, 1057 distribution polygon, 1085 divergence, 334 divergence theorem, 334 divergent infinite products, 346 divergent integral, 301 divergent series, 337 divisibility tests, 3 division -- of asymptotic series, 364 -- of polynomials, 155 -- of segment in given ratio, 82 divisor, greatest common, 4 domain -- boundary, 401 -- closed, 402 -- formula for calculation of volume, 327 -- multiply connected, 409 -- of attraction of fixed point, 877 -- of function, 236 -- on complex plane, 401 -- simply connected, 409 double factorial, 10 double integral, 317 -- change of variables, 321 -- estimation theorem, 318 -- geometric applications, 321 -- geometric meaning, 317 -- mean value theorem, 318 -- physical applications, 321 -- properties, 317 double point, 371 dual linear programming problem, 1017 duality theorem, second, 1018 Duffing equation, 501 Duhamel integral, 516 Duhamel principle, 646 -- first, 646 -- second, 648 dynamic programming, 1027 E eccentricity -- of ellipse, 98 -- of hyperbola, 102 edge -- of dihedral angle, 60 -- of polygon, 51 -- of polyhedral angle, 61 -- of polyhedron, 61 -- of trihedral angle, 60 efficient unbiased estimator, 1089 eigenfunction, 482, 604, 605, 829 -- integral equation, 835 eigenfunctions -- Fredholm integral equation, 829 -- incomplete system, 836 -- system, 836 eigenvalue, 183, 482, 605, 829 eigenvalue problem, 604 -- for linear ordinary differential equations, 482 eigenvalues -- algebraic multiplicity, 209 -- geometric multiplicity, 209 -- linear operator, 211, 209 -- matrix, 183 -- operator, 209 eigenvector -- linear operator, 209 -- matrix, 184 elastic rods, transverse vibration, 1294 electronic mathematical resources, 1451 element -- admissible, 1009 -- conjugate, of group, 231 -- coordinates, 188, 189 -- inverse, 225 -- length, 193 -- neutral, 225 -- norm, 193 -- probability, 1040 -- surface linear, 390 elementary event, 1031 elementary events space, 1031 elementary functions, 19--41 elementary geometry, 43 elementary matrices, 181 elementary symmetric functions, 163 elementary transformation -- of matrix, 180, 199 -- of system of linear equations, 199 elements -- linearly dependent, 188 -- linearly independent, 188 -- of linear space, 187 -- orthogonal, 193 ellipse, 98 -- canonical equation, 98 -- center, 99 -- conjugate diameters, 100 -- diameter, 100 -- directrix, 99 -- eccentricity, 98 -- focal axis, 99 -- focal distance, 98 -- focal parameter, 98 -- focal property, 99 -- focus-directrix property, 100 -- gorge, 145 -- in polar coordinate system, 101 -- left directrix, 99 -- left focus, 99 -- linear eccentricity, 98 -- minor axis, 98 -- numerical eccentricity, 98 -- optical property, 100 -- parameter, 98 -- right directrix, 99 -- right focus, 99 -- semimajor axis, 98 -- semiminor axis, 98 -- throat, 145 -- vertex, 99 ellipsoid, 143, 224 -- imaginary, 224 -- right, 145 -- scalene, 144 -- semiaxis, 143 -- triaxial, 144 elliptic coordinates, 1202 elliptic equation, 585, 590, 642, 972, 1284 -- canonical form, 588 -- first boundary value problem, 633, 643, 644 -- fundamental solution, 642 -- mixed boundary value problem, 644 -- second boundary value problem, 633, 643, 644 -- third boundary value problem, 643, 644 elliptic equations -- boundary value problems, 631, 643, 644 -- linear, 588, 609--611, 631--633, 637, 1284--1298 -- nonlinear, 653, 654, 318--1334 elliptic integral, 969 -- complete, 969 -- first kind, 970 -- incomplete, 970 -- second kind, 970 -- third kind, 971 elliptic modulus, 971 elliptic paraboloid, 145 elliptic point, 394 Emden--Fowler equation, 1223 empirical distribution -- function, 1084 -- variance, 1088 energy conservation law, 996 entire axis, 825, 826, 841 entire function, 414 entry, pivot, 1016 envelope, 377 -- of family of curves, 377 equal complex numbers, 399 equal matrices, 168 equal polynomials, 155 equation -- Abel differential, first kind, 462 -- Abel differential, first kind, canonical form, 463 -- Abel differential, second kind, 464 -- Abel differential, second kind, canonical form, 465 -- Abel functional, 914 -- Abel integral, 1385 -- Abel integral, generalized, 806, 1386 -- Abel integral, second kind, 1392 -- Abel integral, second kind, generalized, 1393 -- Airy, 955 -- algebraic, 162 -- algebraic, binomial, 161 -- algebraic, fourth-degree, 159 -- algebraic, linear, 157 -- algebraic, nth-degree, 162 -- algebraic, quadratic, 158 -- algebraic, reciprocal, 160, 161 -- algebraic, reciprocal, modified, 160 -- autonomous, 1207, 1223, 1225 -- axisymmetric steady hydrodynamic boundary layer, 712 -- Babbage, 917, 1416, 1423 -- Bernoulli, 1207 -- Bessel, 1216 -- Bessel, modified, 1216 -- biharmonic, 1297 -- biharmonic, homogeneous, 1297 -- biharmonic, nonhomogeneous, 1298 -- bilinear functional, 930, 1446 -- binomial algebraic, 161 -- biquadratic, 159 -- Born--Infeld, 659 -- Boussinesq, 709, 1332 -- Burgers, 664, 666, 751, 1307 -- Burgers, radial symmetric case, 1307 -- Burgers--Korteweg--de Vries, generalized, 709 -- Cantor first, 1442 -- Cantor second, 1449 -- Carleman, 1399 -- Cauchy, 1438 -- Cauchy exponential, 1443 -- Cauchy logarithmic, 1439 -- Cauchy power, 1444 -- characteristic, 514, 654 -- characteristic, for difference 881 -- characteristic, for system of ODEs, 529 -- characteristic, of linear operator, 211 -- characteristic, of matrix, 185 -- characteristic, of quadratic form, 150 -- circle, canonical, 97 -- circle, parametric, 98 -- Clairaut, 572, 1262 -- complete, of plane, 124 -- continuity, 784 -- convolution-type, of first kind, 825 -- convolution-type, of second kind, 829 -- cubic, complete, 159 -- cubic, incomplete, 158 -- curve, on plane, 84 -- cyclic functional, 920 -- D'Alembert, 1441 -- damped vibrations, 1213 -- Darboux, 458 -- difference, see difference equation -- differential, see differential equation -- diffusion, linear, see heat equation -- diffusion, nonlinear, with cubic source, 753 -- diffusion, stationary anisotropic, 1323 -- diffusion boundary layer, 1276 -- dispersion, 532, 596 -- Duffing, 501 -- ellipse, canonical, 98 -- elliptic, 585, 588, 590, 1284 -- Emden--Fowler, 1223 -- Ermakov, 1224 -- Euler, 473, 994, 1216 -- Euler--Darboux, 602 -- Euler--Lagrange, 767 -- Euler--Poisson, 1006 -- evolution, second-order, 739 -- exact, first-order ODE, 458 -- exact, second-order ODE, 491 -- family, 377 -- Fisher, 1301 -- Fitzhugh--Nagumo, 1302 -- Fredholm, of first kind, 825 -- Fredholm, of second kind, 829 -- free oscillations, 1213 -- functional, see functional equation -- functional differential, 682, 684, 688 -- gas dynamics model, 558 -- Gauss, 1445 -- Gauss, generalized, 1445 -- Gaussian hypergeometric, 960, 1218 -- Gel'fand--Levitan--Marchenko integral, 1328, 1333 -- general, of plane, 124 -- general, of second-order hypersurface, 220 -- general, of straight line in space, 132 -- general, of straight line on plane, 90 -- Hamilton--Jacobi, 576 -- heat, 585, 672, 680, 698, 728, 1267 -- heat, nonhomogeneous, 1268 -- heat, stationary, with nonlinear source, 1320 -- heat, stationary anisotropic, 1323 -- Helmholtz, 1289 -- Helmholtz, three-dimensional, 600 -- Helmholtz, two-dimensional, 599 -- homogeneous, 591, 592, 810, 829, 834, 1207, 1224 -- homogeneous, generalized, 1210, 1224 -- homogeneous function, 1441 -- hyperbola, canonical, 101 -- hyperbolic, 585, 587, 590 -- hyperbolic, first canonical form, 587 -- hyperbolic, normal, 590 -- hyperbolic, second canonical form, 587 -- hypergeometric, 960, 1218 -- hypergeometric, degenerate, 1215 -- incomplete, of plane, 124 -- information theory, basic, 1440 -- integral, see integral equation -- intercept, of plane, 125 -- intercept-intercept, of straight line, 91 -- involutory functions, 1423 -- Jacobi, 998, 1004, 1007 -- Jensen, 1438 -- Jensen, generalized, 1449 -- 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 -- Kolmogorov, backward, 1076 -- Kolmogorov, forward, 1076 -- Korteweg--de Vries, 752, 756, 759, 1327 -- Korteweg--de Vries, cylindrical, 1328 -- Korteweg--de Vries, modified, 1328 -- L'Huiller, 74 -- Lalesco--Picard, 1403 -- Laplace, ODE, 520 -- Laplace, three-dimensional, PDE, 643 -- Laplace, two-dimensional, PDE, 585, 633, 1284 -- Legendre, 1217--1219 -- Lienard, 1225 -- light rays, 1259 -- likelihood, 1092 -- line, in projections, 133 -- linear, algebraic, 157 -- linear, constant-coefficient, ODE, 473, 514, 1213 -- linear, constant-coefficient, PDE, 588 -- linear, first-order, ODE, 457 -- linear, integral, 801, 810, 824, 829 -- linear, nonhomogeneous, ODE, 473 -- linear, ODE, 472, 518, 1213 -- linear, PDE, 585, 591, 592 -- Liouville, 664 -- Lobachevsky, 1444 -- logarithmic Cauchy, 1439 -- logistic, 875 -- Mathieu, 980, 1221 -- Mathieu, modified, 982, 1220 -- minimal surfaces, 770 -- model, of gas dynamics, 1254 -- Monge--Ampere, 1326 -- Monge--Ampere, homogeneous, 668, 1326 -- Monge--Ampere, nonhomogeneous, 1326 -- Newell--Whitehead, 733 -- noncentral hypersurface, canonical, 224 -- normal, 252 -- normal to surface, 153 -- normalized, of plane, 126 -- normalized, of straight line, 91 -- on entire axis, 825, 826, 841 -- on semiaxis, 826, 842 -- one-dimensional vibration, with central symmetry, 1281 -- ordinary differential, see ordinary differential equation -- Painleve, 495 -- parabola, canonical, 104 -- parabolic, 585, 586, 590 -- parabolic, in narrow sense, 590 -- parametric, of curve on plane, 87 -- parametric, of line in space, 131 -- parametric, of plane, 125 -- partial differential, see partial differential equation -- pencil of planes, 131 -- pencil of straight lines, 92 -- Pexider, 925, 1440 -- Pfaffian, 772, 773 -- plane passing through line of intersection of planes, 131 -- plane passing through point and parallel to another plane, 127 -- plane passing through point and parallel to two straight lines, 129 -- plane passing through three points, 127 -- plane passing through two points and parallel to line, 128 -- Poisson, 1285 -- Poisson, three-dimensional, 599 -- Poisson, two-dimensional, 599 -- quadratic, 158 -- reciprocal, algebraic, 160, 161 -- reciprocal, algebraic, generalized, 160 -- reciprocal, algebraic, modified, 160 -- reciprocal, functional, 919 -- renewal, 1395 -- Riccati, 460 -- Riccati, general, differential, 461, 462, 1207, 1210 -- Riccati, special, differential, 1208 -- Riccati difference, 875, 918, 1428, 1429 -- right-hand side, 801, 825, 829 -- root, 162 -- Schlomilch, 1400 -- Schroder--Koenigs, 911 -- Schrodinger, linear, 1276 -- Schrodinger, nonlinear, 759, 1309 -- Schrodinger, of general form, 1311 -- Schrodinger, with cubic nonlinearity, 1309 -- Schrodinger, with power-law nonlinearity, 1310 -- self-similar solutions, 1442 -- separable, 456, 1207 -- separated, 456 -- sine-Gordon, 1314 -- sinh-Gordon, 1314 -- skew self-distributivity, 1449 -- slope-intercept, of straight line, 89 -- steady transonic gas flow, 1324 -- straight line in polar coordinates, 92 -- straight line in projections in space, 133 -- straight line in space, canonical, 132 -- straight line passing through given point on plane, canonical, 90 -- straight line passing through point and perpendicular to plane, 134 -- straight line passing through two given points, 90, 134 -- straight line passing through two given points on plane, canonical, 90 -- surface, 386 -- surface in space, 119 -- tangent line, 251 -- tangent plane, 268 -- tangent plane to surface, 152 -- telegraph, 1284 -- telegraph, generalized, 626 -- thermal boundary layer, 1276 -- transonic gas flow, 1324 -- transposed, 825, 829, 834 -- traveling wave, 1441 -- Tricomi integral, 1403 -- ultrahyperbolic, 590 -- Urysohn, with degenerate kernel, 863, 864 -- Urysohn integral, nonlinear, 856 -- vibration of string, 1278 -- vibration with axial symmetry, 1282 -- Volterra, with quadratic nonlinearity, 856 -- Volterra integral, nonlinear, 856, 858--861 -- Volterra integral, of first kind, 801, 807 -- Volterra integral, of second kind, 807, 810 -- wave, 585, 1278 -- wave, nonlinear, 723, 1312--1317 -- wave, three-dimensional, 618 -- wave, two-dimensional, 618 -- Whittaker, 960 -- Wiener--Hopf, of first kind, 825, 1401 -- Wiener--Hopf, of second kind, 829, 1406 -- with difference kernel, integral, 825, 829 -- with quadratic nonlinearity, 865 -- with weak singularity of second kind, 829 equations -- admitting variational formulation, 767 -- autonomous, 525 -- Bellman, 1028 -- center of hypersurface, 222 -- difference, 871--907 -- elliptic, linear, 588, 609--611, 631--633, 637, 1284--1298 -- elliptic, nonlinear, 653, 654, 318--1334 -- Fredholm integral, of second kind, 854 -- functional, linear, 907--917 -- hyperbolic, linear, 587, 617, 623--630, 636, 647, 648, 1278--1284 -- hyperbolic, nonlinear, 653, 654, 1312--1317 -- integral, 801--871, 1385--1406 -- integral, linear, 801--855, 1385--1406 -- integral, linear, of second kind, with constant limits of integration, 824, 829 -- integral, nonlinear, 856--871 -- intrinsic, of curve, 388 -- motion of mass point, 87 -- ordinary differential, 453--550, 1207--1228 -- Painleve, 495 -- parabolic, linear, 586, 615--616, 618--622, 634, 635, 646, 648, 1267--1278 -- parabolic, nonlinear, 653, 654, 1301--1311 -- partial differential, 553--798, 1247--1382 -- pencil, 92 -- reaction-diffusion type, 796 -- straight line on plane parametric, 91 -- straight lines on plane, 89 equilateral hyperbola, 101 equilateral triangle, 43, 50, 51 equilibrium point of autonomous system, 545 equilibrium points of two-dimensional linear systems, 534 equilibrium solution, 874, 884 equivalence transformation, 180 equivalent events, 1031 equivalent infinitely small functions, 243 equivalent matrices, 180 equivalent representations of group, 230 Ermakov convergence criterion for series, 340 Ermakov equation, 1224 error function, xxxi, 939, 957, 1267 essential singularity, 413 estimation theorem -- for double integral, 318 -- for triple integral, 325 estimator, 1088 -- asymptotically unbiased, 1089 -- consistent, 1089 -- interval, 1088, 1093 -- maximum likelihood, 1092 -- minimality of variance, 1090 -- statistical, 1088 -- unbiased, 1089 Euclidean space, 192 -- isomorphic, 194 -- normed, 193 Euler constant, xxx, 944, 949, 958 Euler differential equations, 998 Euler equation, calculus of variations, 994 Euler equation, ODE, 473, 520, 1216 Euler formula, 30, 405, 944 Euler line, 47 Euler numbers, 939 Euler polynomials, 989 Euler substitutions, 280 Euler system of ODEs, 542 Euler theorem, 61, 393 Euler transform of first kind, 449 Euler transform of second kind, 450 Euler transformation, 661 Euler triangles, 71 Euler--Darboux equation, 602 Euler--Fourier formulas, 358 Euler--Lagrange equation, 767 Euler--Lagrange theorem, 1009 Euler--Poisson equation, 1006 even function, 236 even number, 3 even permutation, 178 event, 1031 -- complementary, 1031 -- elementary, 1031 -- favorable, 1031 -- frequency, 1083 -- impossible, 1031 -- random, 1031 -- relative frequency, 1083 -- sure, 1031 events -- difference, 1031 -- equivalent, 1031 -- incompatible, 1031 -- intersection, 1031 -- statistically independent, 1036 -- union, 1031 evolute, 378 evolutionary conditions, 792 evolvent, 378, 379 exact differential equation, 458 exact solutions of linear PDEs, 594 example -- Hilbert, 997 -- Weierstrass, 997 excenter, 47 excess, spherical, 72 excess coefficient, 1045 excircle of triangle, 47 excircles, 46 existence and uniqueness theorems -- for ODEs, 454, 488, 524, 543 -- for PDEs, 557, 577 expansion -- asymptotic, 363 -- asymptotic two-term, 478 -- of polynomial in powers of linear binomial, 156 expectation, 1042 -- conditional, 1066 -- of bivariate random variable, 1060 -- of continuous random variable, 1042 -- of discrete random variable, 1042 -- of function of multivariate random variable, 1065 -- population, 1087 expected value of random variable, 1042 exponent, Holder, 310 exponential distribution, 1052 exponential form of complex number, 405 exponential function, 21 -- graphs, 21 -- properties, 22 exponential integral, 940, 958 exponential self-similar solution, 675 extended complex plane, 402 external angle, 43 -- of polygon, 51 extrema, relative, 991 extremal, 994, 998 -- admissible, 994, 998 -- broken, 997, 999 -- nonsingular, 995 -- regular, 995 -- regular point, 995 extremal points of function, 269 extremal properties of quadratic form, 220 extremum -- absolute, 991 -- first-order necessary condition, 994 -- necessary condition, 269 -- point, 269 -- sufficient conditions, 269 extrinsic curvature, 394 F face -- of dihedral angle, 60 -- of polyhedral angle, 60 -- of polyhedron, 61 -- of trihedral angle, 60 factor, reflection, 760 factor group, 227 factorial, xxxi, 10 -- double, xxxi, 10 faithful representation of group, 230 false negative rate, 1096 false position method, 262 false positive rate, 1095 family -- of curves, 377 -- of planes, 150 family equation, 377 favorable event, 1031 feasible region, 1012 feasible solution, 1012 Feller theorem, 1077 Ferrari solution, 161 Feuerbach points, 49 Feuerbach theorem, 49 fifth Painleve transcendents, 499 finite differences, 884 finite games, 1024 finite matrix, 167 finite series, 12 finite subgroup, 228 finite sums, 1111 finite-dimensional Euclidean space, 193 first boundary value problem -- for elliptic equations, 633 -- for hyperbolic equations, 627 -- for ODEs, 480 -- for PDEs, 593 first comparison criterion, 338 first duality theorem, 1018 first group of derivation formulas, 396 first integral, 542, 544, 574 -- for system of ODEs, 544 first mean value theorem, 295 first noteworthy limit, 241 first Painleve transcendent, 496 first quadratic form of surface, 390 first variation of functional, 992 first-order differential constraint, 739 first-order hyperbolic system of quasilinear equations, 780 first-order linear difference equations, 1407 first-order necessary condition for extremum, 994 first-order neighborhood, 991 first-order ordinary differential equation -- Bernoulli, 458 -- linear, 457 first-order ordinary differential equations, 453--472, 1207--1211 -- Cauchy problem, 454 -- homogeneous, 456 -- homogeneous, generalized, 457 -- integrable by quadrature, 453 -- not solved for derivative, 454 -- singular solutions, 455 -- solved for derivative, 453 -- uniqueness and existence theorems, 454 -- with separated or separable variables, 456 first-order partial differential equations, 553, 1247 -- Cauchy problem, 556 -- general solution, 553 -- linear, 1247 -- method of separation of variables, 571 -- nonlinear, 1256 -- quasilinear, 553, 1252 Fisher equation, 1301 Fisher information, 1089 Fisher test, 1104 Fisher transformation, 1105 Fisher--Snedecor distribution, 1086 Fitzhugh--Nagumo equation, 1302 fixed point of function, 908 flat plate -- coordinates of center of mass, 324 -- moments of inertia, 324 flux of vector field, 333 focal axis -- of ellipse, 99 -- axis, of hyperbola, 101 focal distance -- of ellipse, 98 -- of hyperbola, 102 -- of parabola, 105 focal parameter -- of ellipse, 98 -- of hyperbola, 102 focal properties -- of hyperbola, 102 -- of parabola, 105 focal property of ellipse, 99 focal radius -- of parabola, 105 -- right, 99, 102 focus -- of hyperbola, 102 -- of parabola, 105 -- stable, for ODEs, 537 -- unstable, for ODEs, 537 focus-directrix property -- of ellipse, 100 -- of hyperbola, 103 -- of parabola, 105 Folium of Descartes, 374 form -- bilinear, 214, 992 -- bilinear degenerate, 215 -- bilinear nondegenerate, 215 -- bilinear skew-symmetric, 214 -- bilinear symmetric, 214 -- canonical, 462, 473 -- canonical for central surfaces, 143 -- canonical of gas dynamics systems, 786 -- canonical of matrix, 181 -- canonical of quadric, 148 -- canonical of second-order curve, 107 -- linear, 213 -- multilinear, 215 -- multilinear, skew-symmetric, 215 -- multilinear, symmetric, 215 -- normal, 473 -- quadratic, 216, 218, 840 -- quadratic, degenerate, 216 -- quadratic, negative definite, 216 -- quadratic, nondegenerate, 216 -- quadratic, nonnegative, 216 -- quadratic, nonpositive, 216 -- quadratic, positive definite, 216 -- quadratic, second of surface, 392 -- sesquilinear, 213 formal solution, 594 formula -- Bayes, 1036 -- Bernoulli, 1037 -- Bessel, 950 -- Burman--Lagrange, 356 -- Cauchy for ODE, 519 -- Cauchy integral, 409 -- Cauchy multiple integration, 289 -- Cauchy series multiplication, 343 -- Cauchy--Hadamard, 411 -- Chebyshev, 808 -- D'Alembert, 1278 -- de Moivre, 400 -- Dini, 429 -- Dirichlet, 295 -- Euler, 405, 944 -- for calculation of volume of domain, 327 -- Fourier cosine inversion, 445 -- Fourier inversion, 444 -- Frenet, 377 -- Gauss, 396, 808 -- Green, 428 -- Heron, 48 -- Hopf, 565 -- Kontorovich--Lebedev inversion, 448 -- Leibnitz, 256 -- Liouville, 473, 518 -- Liouville for system of ODEs, 540 -- Maclaurin, 257 -- Meijer inversion, 447 -- Newton--Leibniz, 288 -- Newton--Leibniz for function of complex variable, 408 -- of finite increments, 254 -- of integration by parts, 275, 409 -- of repeated integration by parts, 275 -- Peterson--Codazzi, 396 -- Poisson, 949, 1038 -- Post--Widder, 441 -- quadrature, 808 -- Serret--Frenet, 386 -- Stirling, 944 -- Sturges, 1083 -- total probability, 1036 -- Watson, 292 -- Willier, 74 formulas -- Bayes, 1063 -- D'Alembert (Gauss), 73 -- Euler--Fourier, 358 -- Frenet, 377 -- Frobenius, 174 -- Mollweide, 44 -- Peterson--Codazzi, 397 -- Serret--Frenet, 386 -- Weingarten, 396 forward Kolmogorov equation, 1076 Fourier coefficients, 358 Fourier cosine inversion formula, 445 Fourier cosine transform, 445, 449 -- asymmetric, 445 -- tables, 1177 Fourier expansion -- of even functions, 360 -- of odd functions, 360 Fourier inversion formula, 444 Fourier method, 602 Fourier series, 357, 358 -- criteria of uniform convergence, 361 -- differentiation, 359 -- Dirichlet--Jordan convergence criteria, 358 -- in complex form, 360 -- integration, 359 -- Lipschitz convergence criteria, 358 -- of square-integrable functions, 361 -- Riemann localization principle, 358 Fourier sine transform, 445, 446, 449 -- asymmetric, 446 -- tables, 1182 Fourier transform, 443, 449, 614, 843 -- alternative, 444 -- asymmetric, 444 -- asymmetric form, 444 -- inverse, 443 -- n-dimensional, 445 -- solving linear problems of mathematical physics, 615 -- standard form, 443 fourth Painleve transcendent, 498 fourth-degree algebraic equation, 159 fourth-order linear PDEs, 1294 fourth-order nonlinear PDEs, 1332 fraction, 6 fractional integral, 449 fractional power series, 415 Fredholm alternative, 834 -- for symmetric integral equations, 839 Fredholm integral equation -- first kind, 824, 825 -- second kind, 829, 852 -- second kind, with degenerate kernel, 830 -- second kind, with symmetric kernel, 835 Fredholm kernel, 825, 829 Fredholm theorems, 834 free oscillations, 1213 free terms of linear system of equations, 197 frequency -- cumulative, 1084 -- of event, 1083 -- of variate value, 1083 frequency histogram, 1086 frequency polygon, 1085 Frenet formulas, 377 Fresnel integrals, 304, 942 Frobenius formulas, 174 Frullani integral, 307 frustum -- of cone, 67, 68 -- of pyramid, 64 Fuchs index, 749 function -- admissible, 993, 1002, 1006 -- Airy, 477, 955 -- analytic, 413 -- analytic at point, 403 -- analytic at point at infinity, 403 -- analytic in domain, 403 -- antiperiodic, 885 -- approximation, 508 -- arccosine, xxx, 32 -- arccotangent, xxx, 32 -- arcsine, xxx, 31 -- arctangent, xxx, 32 -- autocorrelation, 1072 -- automorphic, 912 -- Bessel, 947, 1216 -- Bessel, modified, 953 -- Bessel, modified, of first kind, xxxi, 953, 1216 -- Bessel, modified, of second kind, xxxi, 953, 1216 -- Bessel, of first kind, xxxi -- Bessel, of second kind, xxxii -- beta, 945 -- boundary, 419 -- bounded, 236 -- bounded variation, 246, 247, 313 -- branch point, 404, 405 -- central cross-correlation, 1073 -- characteristic, 1046, 1066 -- Chebyshev, 985 -- complementary error, xxxi, 939, 958 -- complex variable, 401, 402, 407, 408 -- composite, 237, 265 -- concave, 245 -- continuous, 243 -- convex, 245 -- correlation, 1072 -- cosine, 24, 25 -- cosine, hyperbolic, xxx, 35 -- cotangent, xxx, 24, 26 -- cotangent, hyperbolic, xxx, 35 -- cross-correlation, central, 1073 -- cross-correlation, normalized, 1073 -- cross-covariance, 1072 -- cumulative distribution, 1039 -- cylindrical, 947, 1216 -- decreasing, 236 -- degenerate hypergeometric, xxxii, 956 -- differentiable, 251 -- differentiable at point, 402 -- digamma, 944, 949 -- Dirac delta, 599 -- distribution of two-dimensional random vector, 1057 -- distribution sample, 1084 -- domain, 236 -- elliptic, 972 -- entire, 414 -- entire transcendental, 414 -- error, xxxi, 939, 957, 1267 -- even, 236 -- exponential, 21 -- gamma, xxxii, 943, 1215 -- gamma, incomplete, xxxii, 946, 957, 958 -- Gauss error, xxxi -- Gegenbauer, 1218 -- generating, 815, 1047 -- generating, of numerical sequence, 1047 -- Green, 428, 631, 633--637 -- Hankel, 952 -- harmonic, 404 -- Heaviside unit step, 618 -- Hermite, 986 -- holomorphic at point, 403 -- homogeneous, 1441 -- Hurwitz, 893 -- hyperbolic arccosine, xxx -- hyperbolic arccotangent, xxx -- hyperbolic arcsine, xxx -- hyperbolic arctangent, xxx -- hyperbolic cosine, xxx, 35 -- hyperbolic cotangent, xxx, 35 -- hyperbolic sine, xxxi, 35 -- hyperbolic tangent, xxxi, 35 -- hypergeometric, 960 -- incomplete beta, 947 -- incomplete gamma, xxxii, 946 -- increasing, 236 -- infinitely large, 243 -- infinitely small, 242 -- integrable, 286, 312, 314 -- integrand, 273 -- inverse, 237 -- inverse hyperbolic cosine, 40 -- inverse hyperbolic cotangent, 40 -- inverse hyperbolic functions, 39 -- inverse hyperbolic sine, 39 -- inverse hyperbolic tangent, 40 -- involutory, 917, 1423 -- Jacobi theta, 978 -- joint distribution, 1057, 1064 -- joint probability, 1059, 1064 -- Kummer, 956 -- Lagrange, 270 -- Legendre, 962 -- Legendre, associated, 964, 965 -- Legendre, associated, modified, 966 -- Legendre, associated, of first kind, 965 -- Legendre, associated, of second kind, 965 -- Legendre, of second kind, 1217 -- limit, 264 -- logarithmic, 22 -- Lyapunov, 549 -- Macdonald, 953 -- marginal distribution, 1064 -- marginal probability, 1059 -- Mathieu, 980, 1219 -- Mathieu, modified, 982 -- mean value, 318, 325 -- meromorphic, 414 -- moment-generating, 1047 -- monotone (monotonic), 236 -- multi-valued, 236, 402 -- multivariate random variable, 1065 -- Neumann, 947 -- nonincreasing, 236 -- normalized cross-correlation, 1073 -- odd, 236 -- of exponential growth, 905 -- of several variables, 263 -- of two variables, 263 -- Painleve transcendental, 495 -- parabolic cylinder, 967 -- penalty, 1027 -- periodic, 236 -- Pontryagin, 1011 -- power, 19 -- primitive, 273 -- 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 -- 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 -- Schroder--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 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 Schrodinger 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 -- 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 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 hodograph transformation -- for gas dynamic system, 784 -- for ODEs, 455 -- for PDEs, 657, 658 Holder condition, 310, 430 Holder constant, 310 Holder exponent, 310 Holder inequality, 14, 297, 343 Holder--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--Ampere 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 -- 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 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 -- 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 -- Cramer--Rao, 1089 -- Holder, 14, 297, 343 -- Holder--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 infinitesimal 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 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 -- 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 -- Schlomilch, 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 integration, 829, 1401--1408 -- 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 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 L2-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 Lame coefficients, 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 -- 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 Levy 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 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 integration, 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 -- 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 -- second-order, canonical form, 473 linear partial differential equations, 553--650, 1267--1298 -- second-order, 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 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 convergence 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 -- product, 169 matrix, -- additively inverse, 169 -- adjoint, 170 -- 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 -- 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 -- 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--Poincare, 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 -- 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 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 -- 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--Ampere 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 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--Ampere 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 -- 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 Schrodinger 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 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 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 -- Painleve, 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 Pade approximants, 493 Painleve equations, 495 Painleve test for nonlinear PDEs, 748 Painleve transcendent, 495 -- fifth, 499 -- first, 496 -- fourth, 498 -- second, 496 -- sixth, 499 -- third, 497 Painleve 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 -- 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--Ampere, 1326 -- Monge--Ampere, homogeneous, 668, 1326 -- Monge--Ampere, nonhomogeneous, 1326 -- Newell--Whitehead, 733 -- nonlinear, first-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 -- Schrodinger, linear, 1276 -- Schrodinger, nonlinear, 759, 1309 -- Schrodinger, of general form, 1311 -- Schrodinger, with cubic nonlinearity, 1309 -- Schrodinger, 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 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 Poincare 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 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 -- inscribed in circle, 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 -- 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 -- 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 -- 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 -- first 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 -- 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 -- 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 -- 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 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 -- in Taylor formula in form of Lagrange, 257 -- in Taylor formula in form of Peano, 257 -- in Taylor formula in form of Schlomilch 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, Painleve 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 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 Rouche 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 -- 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 Schlomilch equation, 1400 Schroder--Koenigs functional equation, 911 Schrodinger 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 Painleve transcendent, 496 second partial derivatives, 266 second prolongation of operator, 720 second quadratic form of surface, 392 second variation of functional, 992, 993 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 -- 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 -- 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 Poincare 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--Frenet 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 approximation, 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 Painleve 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 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 -- Levy, 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 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 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, 150 -- 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 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 -- 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 -- 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 -- Rouche, 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 -- Viete, 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 Painleve 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 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, Painleve, 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-Backlund, 663 -- Backlund, 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 -- 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 -- 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 -- characteristic of operator, 209 -- game, 1025 -- of segment, 77 Van der Pol--Krylov--Bogolyubov scheme (method), 503 Vandermonde determinant, 179 Vandermonde matrix, 179 variable -- artificial, 1016 -- basic, 1015 -- incomplete separation, 597 -- nonbasic, 1015 -- phase, 546, 1010 -- random, 1039 -- symmetric random, 1041 variable density, 327 variable parameter, 87 variables -- dependent, 236 -- independent, 236 -- random independent, 1060, 1067 variance, 1044 -- multifactor analysis, 1108 -- of empirical distribution, 1088 -- of random variable, 1044 -- population, 1087 -- residual, 1066, 1108 -- sample, 1088 variate value, frequency, 1083 variation -- coefficient, 1045 -- of argument, 992 -- of functional, 992 -- second of functional, 993 -- total of function, 247 vector, 113, 167, 187 -- associated, 213 -- curvature, 383 -- Darboux, 386 -- geodesic curvature, 392 -- issuing from point A with endpoint B, 189 -- length, 113 -- normal, 124 -- normal curvature, 392 -- normal unit, 389 -- position, 115 -- tangential curvature, 392 -- two-dimensional random, 1057 -- unit, 113 -- zero, 113 vector field -- flux, 333 -- potential, 331 vector opposite to vector, 113 vector space, 187 vector stochastic process, 1072 vectors -- collinear, 113 -- coordinate, 388 -- coplanar, 113 -- cross product, 121 -- difference, 113, 188 -- in space, 113 -- linear combination, 188 -- parallel, 113 -- scalar product, 120 -- scalar triple product, 123 -- sum, 113 -- triple cross product, 122 velocity, characteristic, 790 versatile trigonometric substitution, 282 vertex, 43 -- of ellipse, 99 -- of hyperbola, 102 -- of parabola, 105 -- of polygon, 51 -- of polyhedral angle, 60 -- of polyhedron, 61 -- of second-order curve, 109 -- of trihedral angle, 60 vertical asymptote, 242 vertices of spherical triangle, 71 vibration equation with axial symmetry, 1282 vibration of string, 1276 Viete theorem, 158, 159, 163 viscosity solution, 564 Volterra integral equation -- first kind, 801, 807 -- nonlinear, 856 -- second kind, 807, 810, 858 -- with quadratic nonlinearity, 856 von Mises transformation, 667 von Neumann minimax theorem, 1026 W Wald test, 1097 Watson formula, 292 wave -- breaking, 559 -- centered rarefaction, 558 -- "overturn", 559 -- rarefaction, 559, 785, 1255 -- Riemann, 782, 783 -- shock, 561, 790--793, 1255 wave equation -- linear, 585, 1278 -- linear, three-dimensional, 618 -- linear, two-dimensional, 618 -- nonlinear, 723, 1312--1317 weak convergence, 1069 weak law of large numbers, 1069 weak local maximum, 993, 1001, 1002, 1006 weak maximum of functional, 991 weak neighborhood, 991 weak singularity, 801 -- of second kind, 829 weak solution, 563 Weber parabolic cylinder function, 967 Weber transform, 449 wedge, 63, 65 Weierstrass condition, 998 Weierstrass criterion of uniform convergence, 349 Weierstrass example, 997 Weierstrass function, 998, 1000 -- elliptic, 976 Weierstrass transform, 449 Weierstrass--Erdmann conditions, 997, 999 -- in multidimensional case, 999 weight of variate value, 1083 Weingarten formulas, 396 Whitham rule of "equal areas", 562 Whittaker equation, 960 Whittaker functions, 960 Wiener process, 1086 Wiener--Hopf equation -- first kind, 825, 1401 -- second kind, 829, 1406 Willier formula, 74 Wronskian, 473, 482, 518 -- for system of ODEs, 539 -- of degenerate hypergeometric functions, 958 -- of Legendre functions, 967 Wronskian determinant, see Wronskian Y Y-transform, 448, 449 Z zero -- asymptotic, 364 -- of function, 413 -- simple, 413 zero matrix, 167 zero operator, 204 zero vector, 113 zero-order neighborhood, 991 zeros of Bessel functions, 951 Zhukovskii function, 404, 424 zone, spherical, 69

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