Mathematical, historical, and algorithmic aspects of some classical mathematical con-stants; easy and fast programs are also included.. A directory of mathematics-related Websites ordere
Trang 11452 SOME USEFUL ELECTRONIC MATHEMATICAL RESOURCES
Mathematical Constants and Numbers (http://numbers.computation.free.fr/Constants/constants.
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Trang 2Abel 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
Trang 31454 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
Trang 4axioms 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
Trang 51456 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
Trang 6Cauchy 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
Trang 71458 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