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| Courant R., Hilbert D. — Methods of Mathematical Physics. Volume 1 |
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| Ïðåäìåòíûé óêàçàòåëü |
Infinitely many variables 168 177
Infinitely many variables, applied to integral equations 160
Infinitesimal linear transformations 41—42
Influence function see "Green's function"
inner product of functions 49
Inner product of vectors 1
Integral equation of Abel 158
Integral equations (linear) 112—163
Integral equations (linear) for functions of several variables 152
Integral equations (linear) of the first kind 159
Integral equations (linear) of the second kind 112
Integral equations (linear) of the third kind 161
Integral equations (linear) with discontinuous kernels 152
Integral equations (linear), alternative theorem for 115
Integral equations (linear), Enskog's method for 156
Integral equations (linear), equivalence with differential equations 358—363
Integral equations (linear), Fredholm theorems for 115—122 150
Integral equations (linear), homogeneous 112
Integral equations (linear), inhomogeneous, solution of 136
Integral equations (linear), polar 161
Integral equations (linear), reciprocal 141
Integral equations (linear), singular 154
Integral equations (linear), symmetric 156
Integral equations (linear), transposed 116
Integral equations (linear), Volterra 158 337—339
Integral formula of Dirichlet 78
Integral representations of Bessel functions 474 482
Integral representations of Hankel functions 476
Integral representations of Hermite function 508 509
Integral representations of Laguerre functions 509 510
Integral representations of Legendre functions 502—505
Integral representations of Neumann functions 498
Integral representations of Tchebycheff functions 507—508
Integral theorem for Bessel functions 340—341 490
Integral theorem of Fourier 78
Integral transformations of Mellin 104
Integral transformations, applied to differential equations 466—468
Integral transformations, definition of 112
Integral transforms 113
Integro-differential equations 404
Invariance of Euler equations 222
Invariant variational problems 262
Involutory transformation of variational problems 233
Isoperimetric problem(s) 171—172 216
Isoperimetric problem(s) for plane curves 174
Isoperimetric problem(s) for polygons 166
Isoperimetric problem(s), Euler equation of 207
Isoperimetric problem(s), Hurwitz solution of 97
Isoperimetric problem(s), reciprocity for 258
Iterated bilinear form 12
Iterated kernels 137
Jacobi polynomials 90—91 327—328
Jacobi's method of multiplicative variation 458
Jump conditions 404
Kellogg's method for eigenfunctions 156
Kernels, definite 123 162
Kernels, definition of 112
Kernels, degenerate 114—118 129
Kernels, discontinuous 152
Kernels, eigenfunctions of 113
Kernels, eigenvalues of 113 132 137
Kernels, expansion theorem for 134 140
Kernels, functions of 157
Kernels, iterated 137
Kernels, positive definite 132
Kernels, possessing no eigenvalues 157 158
Kernels, reciprocal 141
Kernels, resolvents of 141—147 162
Kernels, symbolic functions of 157
Kernels, symmetrizable 161
Kernels, unsymmetric, adjoint orthogonal systems of 159
kinetic energy 242
Lagrange multipliers 165
Lagrange's equations of motion 243
Laguerre functions 509—510
Laguerre orthogonal functions 154
Laguerre polynomials 93—97 329—330 342—343
Laguerre polynomials, completeness of 95
Laguerre polynomials, generating function for 510
Laguerre's equation, Green's function for 375—376
Lame equation 320
Lame functions 320
Lame problem 319—324
Laplace integral 82
Laplace spherical harmonics 316 510—521
Laplace transformation 467
Laplace's equation see "Potential equation"
Laplace's integral representation for Legendre functions 503—504
Laplacian, transformation to polar coordinates 224
Least time, principle of 165
Lebesgue convergence theorem 109—111
Lebesgue integral 108—111
Legendre condition 186 200 202 214
Legendre condition, strong form of 216
Legendre functions 325—327 501—507
Legendre functions of the second kind 504—505
Legendre functions, associated 327 505
Legendre functions, integral representations for 502—505
Legendre polynomials 82—87 402
Legendre polynomials as special spherical harmonics 318
Legendre polynomials of higher order 325 505
Legendre polynomials, asymptotic expansion of 533—535
Legendre polynomials, completeness of 82
Legendre polynomials, differential equation for 86
Legendre polynomials, generating function for 85
Legendre polynomials, minimum property of 86
Legendre polynomials, recursion formula for 86
Light rays 165 170 189 213 258
Linear dependence of functions 50
Linear dependence of vectors 2
Linear differential operator 275
Linear equations 5—6
Linear equations, perturbations of 42—44
Linear equations, solution by means of eigenvectors 30
Linear transformations 1—47
Linear transformations, infinitesimal 41—42
Mathematical physics, problems of 275—396
Mathieu functions 392
Matrices 7—11
Matrices, characteristic values of 17
Matrices, eigenvalues of 17 26
Matrices, elementary divisors of 45
Matrices, functions of 22
Matrices, spectra of 17
Maximizing sequences 175
Maximum-minimum property of eigenvalues of a differential equation 405
Maximum-minimum property of nodal points 463
Maxwell — Sylvester representation for spherical harmonics 514—521
Mean convergence 98 110
Mean square error 52
Mean, approximation in the 52 54
Measure of a set 108
Measure of independence 61
Mellin transformation 103—105
Membrane 246 249 297—307 463
Membrane, circular 302—306
Membrane, eigenvalues for 298
Membrane, forced motion of 300
Membrane, non-homogeneous 306
Membrane, rectangular 300—302
Mercer's theorem 138—140
Mercer's theorem, application to differential equations 360 363 369
Minimal surfaces 193 258
Minimizing sequences 175
Minimum property of eigenfunctions 161
Minimum property of eigenvalues, derived from completeness 456
Minimum property of Legendre polynomials 86
Minimum-maximum principle, for quadratic forms 31
Minimum-maximum principle, for symmetric kernels 132 134
Momentum, conservation of 265
| Muentz's Theorem 101—102
Multiple Fourier series 73
Multiplicative variation, Jacobi's method of 458
Multiplicity of an eigenvalue 113 129
Multiplier rule for variational problems 222
Multipliers, Lagrange 165
Multipole 515
Natural boundary conditions 208
Neighborhood of a function 169
Neumann functions 471—473 496
Neumann functions, singularities of 501
Neumann series, for integral equations 140—142
Neumann series, of bilinear forms 18
Neumann series, of matrices 10
Nodal lines 300 302 304 395
Nodal points for Sturm — Liouville problem 454 463
nodes 451—154 458
Noether's theorem 262
Norm of a function 49
Norm of a vector 2
Normal coordinates 282
Normal modes 283
Null sets 108
Orthogonal function system, of Hermite 154
Orthogonal function system, of Laguerre 154
Orthogonal group 539
Orthogonal systems of functions 40—50 87
Orthogonal systems of vectors 3—5
Orthogonal transformations 14—16 536
Orthogonal transformations in Hiloert space 55
Orthogonalization of functions 50
Orthogonalization of vectors 4
Orthonormal system of functions 49
Orthonormal system of vectors 3
Oscillation theorem 454
overtones 286
Paraboloidal coordinates 229
Parallelepiped, confocal 319
Parameters in boundary conditions 392—393 461
Particle mechanics, integrals in 265
Perturbation theory 343 350
Perturbations 42
Picard's theorem 160
Piecewise continuous functions 48
Piecewise smooth functions 48
Pitch 288
Plate 250—251 307—308 403 463
Plate, asymptotic behavior of eigenvalues for 46C
Plucked string 388
Point force 351
Poisson's equation 368
Poisson's integral 513—14
Poisson's summation formula 76
Polar form of a quadratic form 12
Polar integra1 equations 161
Polynomials, completeness of 65 101—102
potential energy 242
Potential equation in space, Green's function for 378—384
Potential equation in the plane, Green's function for 384—386
Potential equation, Green's function for 363—371 377
Potential equation, transformation to polar coordinates 224
Potential functions, conjugate 241
Potential theory 178—182 315—324 363—371 377—388
Principal axes transformations, of quadratic forms 23—31
Principal axes transformations, of two quadratic forms simultaneously 37—41
Principle of Castigliano 270
Principle of Fermat 165
Propagation of light 170 189 214
Pythagorean theorem for function spaces 253
Quadratic forms 12—16 23—34
Quadratic forms, constraints on 33 44
Quadratic forms, degenerate 27
Quadratic forms, inertial theorem for 28
Quadratic forms, polar forms of 12
Quadratic forms, simultaneous transformation to principal axes 37—41
Quadratic forms, transformation to principal axes 23—31
Quadratic integral form 122
Quaternions 539
Rayleigh — Ritz method 175
Reciprocal bilinear form 12
Reciprocal bilinear form, Neumann series for 19
Reciprocal eigenvalue 17
Reciprocal kernel 141
Reciprocal spectrum 17
Reciprocal variational problems 235 252—257 270
Reciprocity for isoperimetric problems 258
Reciprocity formulas for Fourier integrals 80
Rectangular parallelepiped, confocal 319
Recursion formula for Legendre polynomials 86
Reflection method of constructing Green's function 378—379 384—385
Reflection, law of 165
Refraction, law of 166
Regions with separate components 405
Rellich, lemma of 414 450
Resolvent of a bilinear form 18 29
Resolvent of a kernel 141—147 162
Resolvent of a symmetric kernel 142
Resonance 285
Riesz — Fischer theorem 110
Ritz see "Rayleigh — Ritz"
Rod 245 295—297
Rod, boundary conditions for 295
Rod, eigenvalues for 296
Saddle point method 526 —533
Schlaefli's integral 501—502
Schmidt method for derivation of Fredholm theorem 155
Schmidt orthogonalization process see "Gram — Schmidt"
Schroedinger eigenvalue problems 445—450
Schroedinger's equation 342—343
Schwarz inequality for functions 49
Schwarz inequality for vectors 2
Second variation 214
Self-adjoint differential expression 279
Separated regions 405
Separation of variables 287 313
Series expansions of arbitrary functions 48—111
Series, Fourier 69—77
Series, Neumann 10 18 140—142
Singular equations, eigenvalues of 461
Singular integral equations 154
Small vibrations 244
Smooth sequences of functions 63
Snell's Law of Refraction 166
Spectral decomposition 99—100
Spectral density 100
Spectrum 339—340
Spectrum of a matrix 17
Spectrum of a unitary matrix 46
Spectrum, continuous 100 339—343
Spectrum, discrete 130
Spectrum, reciprocal 17
Spherical coordinates 225
Spherical harmonics 314 316 342—343 510
Spherical harmonics, completeness of 512—513
Spherical harmonics, expansion theorem for 513
Spherical harmonics, generating function for 539—541
Spherical harmonics, Maxwell — Sylvester representation for 514—521
Spherical harmonics, symmetric 512
Spherical harmonics, transformations of 535—545
Spheroidal coordinates 229
Stationary functions and curves 186
Stationary values in variational problems 186
Steepest descent, method of 526—533
Steiner's problem 166
Stirling's formula 522—524
String 244 388—392 463
String, boundary conditions for 291
String, eigenvalues for 287
String, forced vibrations of 289 389
String, free motion of 287
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