1452 SOME USEFUL ELECTRONIC MATHEM ATICAL RESOURCES Mathematical Constants and Numbers (http://numbers.computation.free.fr/Constants/constants. html). Mathematical, historical, and algorithmic aspects of some classical mathematical con- stants; easy and fast programs are also included. Mathematical WWW Virtual Library (http://www.math.fsu.edu/Virtual/index.php). A directory of mathematics-related Websites ordered by type and mathematical subject. Mathematics Archives (http://archives.math.utk.edu). Combined archive and directory of mathe- matics Websites, mailing lists, and teaching materials. Mathematics Genealogy Project (http://www.genealogy.ams.org). A large biographical database of mathematicians. Mathematics Websites (http://www.math.psu.edu/MathLists/Contents.html). A directory of math- ematics Websites ordered by type and mathematical subject. Math Forum: Internet Mathematics Library (http://mathforum.org). Adirectoryofmathematics Websites ordered by the mathematical subject. MathGuide: SUB Gottingen (http://www.mathguide.de). An Internet-based subject gateway to mathematics-related Websites. MathWorld: World of Mathematics (http://www.mathworld.com). An online encyclopedia of mathematics, focusing on classical mathematics. The Web’s most extensive mathematical resource. MGNet (http://www.mgnet.org/mgnet-codes.html). Free software. Netlib (http://www.netlib.org). A collection of mathematical software, papers, and databases. NumericalSolutions(http://www.numericalmathematics.com/numerical solutions.htm). A library of mathematical programs. PlanetMath.Org (http://planetmath.org). An online mathematics encyclopedia. Probability Web (http://www.mathcs.carleton.edu/probweb/probweb.html). A collection of prob- ability resources on the World Wide Web; the pages are designed to be especially helpful to researchers and teachers. Science Oxygen—Mathematics (http://www.scienceoxygen.com/math.html). Topics from vari- ous sections of mathematics. Scilab (http://scilabsoft.inria.fr). A free scientific software package. Software — Differential Equations (http: / / www.scicomp.uni-erlangen.de / SW / diffequ.html). General resources and methods for ODEs and PDEs. S.O.S. Mathematics (http://www.sosmath.com). A free resource for math review material from algebra to differential equations. Statistics Online Computational Resources (http://socr.stat.ucla.edu). Interactive distributions, statistical analysis, virtual probability-relatedexperiments and demonstrations, computer games, and others. Stat/Math Center (http://www.indiana.edu/ statmath/bysubject/numerics.html). Numerical com- puting resources on the Internet. UW-L Math Calculator (http://www.compute.uwlax.edu/index.php). Calculus, differential equa- tions, numerical methods, statistics, and others. Wikipedia: Free Encyclopedia—List of Open Source Software Packages (http://en.wikipedia. org/wiki/List of open-source software packages). Wikipedia: Free Encyclopedia—Mathematics (http://en.wikipedia.org/wiki/Mathematics). A collection of short articles from various sections of mathematics. Wolfram Functions Site (http://functions.wolfram.com). More than 87,000 formulas used by mathematicians, computer scientists, physicists, and engineers; more than 10,000 graphs and animations of the functions. Yahoo — Mathematics Websites (http://dir.yahoo.com/science/mathematics/). A directory of mathematics Websites ordered by type and mathematical subject. Yahoo—Software (http://dir.yahoo.com/Science/Mathematics/Software). Adirectoryofsoftware. INDEX 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 1453 1454 INDEX 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 Poincar ´ e 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-B ¨ acklund 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 INDEX 1455 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 B ¨ acklund 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 1456 INDEX boundary (continued) 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 B ¨ urman–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 INDEX 1457 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, 151 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 characteristic equation (continued) 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 1458 INDEX classification (continued) 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 Lam ´ e, 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 H ¨ older, 310, 430 initial homogeneous, 591 invariance, 720, 732 invariant surface, 732 Jacobi, 998, 1000, 1004, 1007 Jacobi, strengthened, 1000, 1004, 1007 jump, 561 . 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,. scientists, physicists, and engineers; more than 10,000 graphs and animations of the functions. Yahoo — Mathematics Websites (http://dir.yahoo.com/science /mathematics/ ). A directory of mathematics Websites. (http://www.math.psu.edu/MathLists/Contents.html). A directory of math- ematics Websites ordered by type and mathematical subject. Math Forum: Internet Mathematics Library (http://mathforum.org). Adirectoryofmathematics Websites ordered