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| Ïîèñê ïî óêàçàòåëÿì |
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| Courant R., Hilbert D. — Methods of Mathematical Physics. Volume 1 |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel's integral equation 158
Accumulation principle for functions 57
Adjoint differential expression 279 280
Adjoint orthogonal systems of unsymmetric kernels 159
Alternative theorem for integral equations 115
Alternative theorem for linear equations 6
Alternative theorem for ordinary differential equations 359
Analytic continuation 395
Approximation in the mean 52 54
Approximation theorem of Weierstrass 65
Argument function 168
Arzela's theorem 59
Associated Legendre functions 327 505
Asymptotic behavior of eigenvalues, in Sturm — Liouville problem 414
Asymptotic behavior of eigenvalues, of Bessel's equation 415
Asymptotic behavior of eigenvalues, of vibrating plate 460
Asymptotic behavior of solutions of the Sturm — Liouville problem 331—339
Asymptotic dimension of a sequence of functions 63
Asymptotic distribution of eigenvalues of a differential equation 429—445 460
Asymptotic expansions 522—535
Asymptotic expansions for Bessel and Hankel functions 524 532
Beats 390
Bessel functions 303—306 324—325 340 372 390—391 402 467
Bessel functions, asymptotic behavior of 332—334
Bessel functions, asymptotic representations of 524 532
Bessel functions, functional relation for 489
Bessel functions, integral representations for 474—82
Bessel functions, integral theorem for 340—341 490
Bessel functions, orthogonality relations for 305
Bessel functions, power series for 482—484
Bessel functions, relations between 485
Bessel functions, singularities of 501
Bessel functions, zeros of 451 492—496
Bessel transformation 490—492
Bessel's equation 303 467
Bessel's equation, asymptotic behavior of eigenvalues of 415
Bessel's equation, singularities of solutions of 501
Bessel's inequality for functions 51 425
Bessel's inequality for vectors 5
beta function 483
Bezout's theorem 520
Biharmonic equation 370
Bilinear forms 11—14
Bilinear forms, reciprocal 12
Bilinear forms, resolvents of 18. 29
Bilinear formula, for iterated kernels 137
Bilinear integral form 123
Bilinear relation 360
Biorthogonality relations 404
Boundary conditions for the rod 295
Boundary conditions for the string 291
Boundary conditions, containing a parameter 392—393 461
Boundary conditions, homogeneous and inhomogeneous 277
Boundary conditions, natural 208
Boundary, free 208
Brachistochrone 170 207
Broken extremals 259
Buckling, variational problem of 272
Calculus of perturbations 344
Calculus of variations 164—274
Calculus of variations, applied to eigenvalue problems 397—165
Calculus of variations, Fundamental Lemma of 185
Canonical differential equations 239
Canonical form of variational problems 238
Castigliano's principle 270
Catenary 172 218
Cayley's theorem 22 536—539
Characteristic values of a matrix 17
Closed systems of functions 110
Complete systems of functions 52 54
Completeness of a system of functions 110
Completeness of eigenfunctions of a differential equation 359—360 369 424
Completeness of Hermite polynomials 96—97
Completeness of Laguerre polynomials 96
Completeness of Legendre polynomials 82
Completeness of powers of x 65 101—102
Completeness of spherical harmonics 512—513
Completeness of systems of functions of several variables 56
Completeness of systems of vectors 4
Completeness of trigonometric functions 68
Completeness relation for functions 51
Completeness relation for vectors 5
Components of a function 51
Conduction of heat 311
Confocal rectangular parallelepiped 319
Conformal mapping 377
Conjugate potential functions 241
Conservation of momentum 265
Constrained systems, eigenvalues of 408
Constraints on quadratic forms 33 44
Constraints on variational problems 233
Continued fraction for the cotangent 489
Continuous dependence of eigen values and eigenfunctions on the kernel 151
Continuous spectrum 100 339—343
Convergence in the mean 98 110
Convergence theorems of Lebesgue 109—111
Cotangent, continued fraction for 489
Cylinder functions see "Bessel functions" "Hankel "Mathieu "Neumann
Cylindrical domain 319
Darboux method 532—535
Definite kernels 123 162
Degeneracy of eigenvalues 129
Degenerate kernels 114—118 129
Degenerate quadratic forms 27
Dense function systems 100—101
Derivative, variational 186
Determinant inequality of Hadamard 36
Dido's problem 258
Differential equations, canonical 239
Differential equations, equivalence with integral equations 358—363
Differential equations, integral transformations applied to 466—468
Dimension number, asymptotic 63
Dini's theorem 57
Dipole 514—15
Direct solution of variational problems 174—183
Dirichlet's discontinuous factor 81
Dirichlet's integral formula 78
Dirichlet's problem 240
Dirichlet's problem for the circle 179
Discontinuous factor of Dirichlet 81
Discontinuous kernels 152
Discrete spectrum 130
Divergence expressions 195
du Bois-Reymond, theorem of 200
Eigenfrequency 283
Eigenfunctions for rectangular membrane 301
Eigenfunctions of a differential equation, nodes of 451—455
Eigenfunctions of a differential equation, nodes of, completeness of 424
Eigenfunctions of a kernel 113
Eigenfunctions of a kernel, minimum properties of 161
Eigenfunctions of symmetric kernels 126—132 148—150
Eigenfunctions, continuous dependence on kernel 151
Eigenfunctions, Kellogg's method for 156
Eigenfunctions, nodes of 300 302 304 395 451—455 458 463
Eigenfunctions, zeros of 451—455
Eigenvalue problems for closed surfaces 461
Eigenvalue problems of mathematical physics 275—396
Eigenvalue problems of Schroedinger type 445—450
Eigenvalue problems with two parameters 460
Eigenvalue problems, definition of 309
Eigenvalue problems, variational methods for 397—165
Eigenvalues for the membrane 298
Eigenvalues for the plate, asymptotic behavior of 460
Eigenvalues for the rod 296
Eigenvalues for the string 287
Eigenvalues of a constrained system 408
Eigenvalues of a differential equation for a subdomain 408—409
Eigenvalues of a differential equation, asymptotic distribution of 429—445 460
Eigenvalues of a differential equation, continuity of 418—424
Eigenvalues of a differential equation, dependence on boundary conditions 410—411
| Eigenvalues of a differential equation, dependence on coefficients 411
Eigenvalues of a differential equation, extremum properties of 398—407
Eigenvalues of a differential equation, infinite growth of 412
Eigenvalues of a differential equation, maximum-minimum property of 405
Eigenvalues of a differential equation, minimum properties of 459
Eigenvalues of a kernel 113 132 137
Eigenvalues of a matrix 17 26
Eigenvalues of a positive definite kernel 132
Eigenvalues of a symmetric kernel 148—150
Eigenvalues of a symmetric kernel, first 122—125
Eigenvalues of a symmetric kernel, maximum-minimum property of 132—134
Eigenvalues of a symmetric kernel, totality of 126—132
Eigenvalues of a unitary matrix 46
Eigenvalues of Bessel's equation, asymptotic behavior of 415
Eigenvalues of functions of a kernel 157
Eigenvalues of functions of a matrix 22
Eigenvalues of infinite multiplicity 154 395
Eigenvalues of Sturm — Liouville problem, asymptotic behavior of 414
Eigenvalues, continuous dependence on kernel 151
Eigenvalues, degenerate 129
Eigenvalues, minimum-maximum property 31 132—134 405
Eigenvalues, multiple 113 129
Eigenvalues, negative 294 416
Eigenvalues, reciprocal 17
Eigenvectors 24
Eigenvectors, use in solving linear equations 30
Eigenvibrations 283 310
Eigenvibrations of homogeneous string 287
Elastic bodies 268
Electrostatics 267
Elementary divisors of a matrix 45
Ellipsoidal coordinates 226
Elliptic coordinates 319 391—392
Elliptic functions 227
Energy integral 265
Energy, kinetic 242
Energy, potential 242
Enskog's method for symmetric integral equations 156
Equicontinuous functions 58 114 119 125
Equilibrium conditions of a mechanical system 243
Equilibrium problems 308
Erdmann vertex condition 260
Euler equations 183—207
Euler equations for isoperimetric problem 207
Euler equations, homogeneous form of 196
Euler equations, integration of 206
Euler equations, invariance of 222
Euler expression, identical vanishing of 193
Euler integral 483
Euler transformation 467
Expansion theorem 310
Expansion theorem for definite kernels 138—140
Expansion theorem for differential equations 360—361 427—428
Expansion theorem for iterated kernels 137
Expansion theorem for spherical harmonics 513
Expansion theorem for symmetric integral equations 134—136
Expansion theorem, limits of validity of 395—396
Expansions, asymptotic 522 535
Extrema, Weierstrass's theorem on 20 164
Extremals 185
Extremals, broken 259
Extremum properties of eigenvalues 398—407
Fejer's summation theorem 102
Fermat's principle 165
Finite differences 176
First variation 184 208—214
Fischer — Riesz theorem 110
Forced motion of the membrane 300
Forced motion of the string 289 294
Forced vibrations 285 389
Form factor 286
Forms see "Particular type of form"
Fourier coefficients 51 424
Fourier coefficients, order of magnitude of 74
Fourier integral 77—82 98—100
Fourier integral theorem 78
Fourier series 69—77
Fredholm formulas 142—147
Fredholm's theorems for arbitrary kernels 118—122 150
Fredholm's theorems for degenerate kernels 115—118
Fredholm's theorems, Schmidt's derivation of 155
Free boundaries 208
Free vibrations 282—283
Friedrichs transformation 234
Function space 57
Function space, gradient in 222
Functionals 167—169
Functions of a kernel, eigenvalues of 157
Functions of a matrix, eigenvalues of 22
Fundamental domain 48 112
Fundamental lemma of the calculus of variations 185
Fundamental solution 370
Fundamental solution of an ordinary differential equation 353
Fundamental tone 286
Generalized Green's function 356—357
Generating function for Hermite and Laguerre polynomials 510
Generating function for spherical harmonics 539—541
Geodesic curves 170 189 213
Gibbs' phenomenon 105—107
Gradient in function space 222
Gram determinant of a system of functions 62 107—108
Gram determinant of a system of vectors 34—36
Gram — Schmidt orthogonalization process 4 60
Green's formula 278—280
Green's function 311 351—388
Green's function as kernel of an integral equation 358
Green's function for a circular ring 306—388
Green's function for Hermite's equation 373—375
Green's function for Laguerre's equation 375—376
Green's function for ordinary differential equations 351—363
Green's function for ordinary differential equations of higher order 362—363
Green's function for ordinary differential equations, construction of 354—355
Green's function for ordinary differential equations, definition of 352—353
Green's function for the biharmonic equation 370
Green's function for the potential equation 363—370 377
Green's function for the potential equation in space 378—384
Green's function for the potential equation in the plane 384—386
Green's function in the generalized sense 356—357
Green's function, construction by reflection 378—379 384
Green's function, examples of 371—388
Green's function, obtained by conformal mapping 377
Green's function, symmetry of 354
Green's tensors for systems of differential equations 393—394
Group, orthogonal 539
Haar, theorem of 203
Hadamard's inequality for determinants 36
Hamilton's variational principle 242
Hammerstein's theorem 162
Hankel functions 468—473
Hankel functions, asymptotic representations for 524—532
Hankel functions, integral representation of 476
Hankel functions, singularities of 501
Harmonic functions see "Potential equation"
Heat conduction, equation of 311
Herglotz, theorem of 542
Hermite equation 508
Hermite equation, Green's function for 373—375
Hermite functions 508—509
Hermite orthogonal functions 154
Hermite polynomials 91—93 328
Hermite polynomials, completeness of 96—97
Hermite polynomials, generating function for 510
Hermitian forms 13 28
Hilbert space 55
Hilbert space, orthogonal transformations in 55
Holonomic conditions 221
Hurwitz's solution of the isoperimetric problem 97—98
Independence, measure of 61
Indicatrix 258
Inertial theorem for quadratic forms 28
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