EqWorld The World of Mathematical Equations

## Handbook of Mathematics for Engineers and Scientists

Publisher: Chapman & Hall/CRC Press
Publication Date: 27 November 2006
Number of Pages: 1544

## 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
-- asymptotic series, 364
-- hyperbolic functions, 37
-- inverse hyperbolic functions, 41
-- inverse trigonometric functions, 33
-- matrices, 168
-- real numbers, 6
-- trigonometric functions, 28
additive inverse of real number, 6
additive separable solution, 571, 596, 597, 678
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
-- 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
-- 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
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
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
-- 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--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
-- 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 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
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
-- 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
-- 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
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
-- 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
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
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
-- 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
-- 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, 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
-- 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
-- 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
-- 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 weak singularity of second kind, 829
equations
-- 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
-- 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
-- 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
-- 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 second-order curve, 107
-- linear, 213
-- multilinear, 215
-- multilinear, skew-symmetric, 215
-- multilinear, symmetric, 215
-- normal, 473
-- 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
-- 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
-- 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
-- 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
-- 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
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 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
-- 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 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 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 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 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
-- 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
-- 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
-- rank, 177, 178
-- real, 171
-- rectangular, 177
-- 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
-- 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, 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, for integral equations, 870
-- 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 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
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
-- 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
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 symmetric operator, 207
nonorthogonal polynomials, 988
nonparametric hypothesis, 1094
nonpositive Hermitian operator, 207
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 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
-- 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
-- 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
-- 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
overdetermined systems of PDEs, 770, 771

### P

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
-- 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
-- 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
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, 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
pole, 80, 413
-- order, 413
-- simple, 413
polygon, 51
-- center, 55
-- 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 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
-- 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
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 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

-- 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
-- positive definite, 993
-- strongly positive, 993
-- of functions, 297
quadrature method, for integral equations, 852, 861, 870
-- canonical equations, 143
-- canonical form, 143, 148
-- classification, 148
-- general equation, 148
-- general theory, 148
-- invariant, 148
-- semi-invariant, 148
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
-- 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
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 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

-- 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
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-intersection, point, 372
self-similar solution, 669, 1442
-- continuous, 781
-- exponential, 675
-- generalized, 677
-- limiting, 675
semi-invariant
-- for probability distribution, 1046
-- 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
-- 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
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
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
-- 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
-- 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 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
-- 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
-- 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
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
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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