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Miller W. — Symmetry and Separation of Variables

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Название: Symmetry and Separation of Variables

Автор: Miller W.

Аннотация:

This volume is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner. Some modern group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory are shown. In particular, it is shown explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory.

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Рубрика: Математика/Симметрия и группы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1977

Количество страниц: 285

Добавлена в каталог: 30.04.2005

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Предметный указатель

Addition theorem      35 37 57 176 202
Adjoint representation      12 20 41 59 77 107 124 163 243
Airy function      89 138
Angular momentum operator      243
Anharmonic oscillator function      127 138 147
Appell transform      94 146 154
Bargmann — Segal Hilbert space      142 232
Basis      26 263
Bessel equation      10 28 43 62 270
Bessel function      10 56 62 119 136 152 165 194 216 237 239 243 269—271
Bessel function of second kind      72
Bessel function, modified      201 221
Bessel function, spherical      175 190
Bessel polynomials      72
Bilinear expansion      39 71 159 193 222
Binomial coefficient      266
Bipolar coordinates      212
Cartesian coordinates      9 17 43 50 177 207 234
Casimir operator      86 110 213 236 239 243
Cauchy problem      96 151 159
Cauchy — Riemann equations      14
Cherry's thoerem      89
Class I equation      8 41 157 162
Class II equation      8 157
Clebsch — Gordan coefficient      175
Commutator      261 263
Commutator, matrix      5 261
Confluent hypergeometric function      119 140 152 153 155 167 258 268 269
Conformal algebra      224 240 242
Conformal group      205 228 230 233 242
Conical coordinates      190 192 208
Continuous spectrum      30
Cyclide      208 209 212
Cylindrical coordinates      165 177 192 196 207 212 221
Cylindrical wave      36 177 202 216
Deficiency indices      51 53 54
Dirac delta function      22 29
Eigenvalue      25
Eigenvector      25
Ellipsoidal coordinates      163 182 192 208
Ellipsoidal harmonic      208
Ellipsoidal wave equation      168 182
Ellipsoidal wave function      168 182 183
Elliptic coordinates      18 134 182 185 186 190 234
Elliptic cylinder coordinates      178 179 192 207
Elliptic function      168 182 211 274 275
Equivalent coordinates      12 230
Euclidean group      2 161
Euclidean group in the plane      2—6 59 234
Euclidean group in three space      161 212
Euclidean group, complex      59
Euler angles      161 172
Euler — Poisson — Darboux equation      239—241 243
Exponential mapping      219 261 263
extension of operator      26
Fourier series      27
Fourier transform      50 225
Galilei group      75 79 128 147
Gamma function      88 265
Gegenbauer polynomial      199 219 220 222 239 241 243 267
Generalized eigenfunction      29
Generalized hypergeometric function $_{p}F_{q}$       156 258 259 271 272
Generating function      63 115 154 155 199 220 222 256 257 259
Generating function, continuous      90 112 159
Graf's addition theorem      68 71
Hankel function      57 195
Hankel transform      112
Heat equation      92
Heat equation in three variables      145
Heat equation in two variables      92
Heat equation, complex      97 151 222
Heat group      93 146
Heat group, complex      151
Heat polynomials      94
Helmholtz equation      1 22 151 160 165 171 191 222 234
Helmholtz equation, complex      58
Hermite function      98 102 156 159
Hermite polynomial      84 98 99 131 270
Hilbert space      22 23 47 51 80 128 169 171 226 231 235 236 238 240 242
Hille — Hardy formula      110 121 159
Hypergeometric function $_{2}F_{1}$       222 241 245 250 252 266—268
Ince polynomial      135 142 190
Inversion symmetry      205 225
Jacobi polynomial      241 243 268
Kepler problem      231 242
Klein — Gordon equation      39 235 242
Laguerre function      104 113
Laguerre polynomial      101 109 119 134 149 152 198 231 257 269
Lame equation      169 186 188 208 211 218 233
Lame polynomial      169 184 187—190 208 237
Lame — Wangerin function      237 239
Laplace equation      204 219 222
Laplace operator      11
Laplace operator on the hyperboloid      236 242
Laplace operator on the sphere      174 184 230 233 242 243
Lauricella function      245—258 272
Lebedev transform      53
Legendre function      165 166 173 179 195 212 237 239 267 268
Lie algebra      3 4 58 261
Lie derivative      262
Lie group      260
Limit in the mean      47 49
Local group representation      65 262
Lorentz group      242
Lorentz transformation      223
Macdonald function      51 53 56 57 237
Mathieu equation      19 33 46 178 272
Mathieu function      19 44 45 137 143 192 208 272 273
Mathieu function, modified      34 137 179
Matrix elements      35 56 66 91 100 103 117 173 175 198 200 252
Matrix elements, mixed-basis      36 91 112 237

Mehler's theorem      102 103
Mellin transform      88
Multiplicity of spectrum      30
Multiplier      262
Multiplier, representation      262
Nonsplit coordinates      230 241
Oblate spheroidal coordinates      166 180 192 207 216
Operational identity      202
Orbit, in Lie algebra      12 21 41 59 78 79 82 93 97 107 109 113 124 163 222
Orthogonal group O(3)      185
Orthogonal vectors      25
Orthonormal set      26
Overlap functions      35 55 90 112 142 190 191
Parabolic coordinates      17 33 69 137 167 180 191 197 203 208 215
Parabolic cylinder coordinates      178 192 207 222 234
Parabolic cylinder equation      17 33 270
Parabolic cylinder function      17 33 43 69 88 98 139 269 270
Paraboloidal coordinates      167 181 208
Parseval's equality      50
Planck's constant      73
Plane wave      36 177
Pochhammer's symbol      266
Poincare group      39 40 59 224 235 242
Poincare group, extended      42 55
Polar coordinates      10 18 62 63 136 137 153 234
Potential function      73 121 157
Prolate spheroidal coordinates      166 179 192 207 222
R-separable coordinates      78 125 130 204 230 259
R-separable coordinates for heat equation      94 97 146—149 151
R-separable coordinates for Laplace equation      209 210
R-separable coordinates for radial free particle Schroedinger equation      107 113
R-separable coordinates for three-variable Schroedinger equation      126
R-separable coordinates for two-variable Schroedinger equation      79
R-separable coordinates for wave equation      230 232—240
Raising operator      158
Real forms of Lie algebra      59
Recurrence formulas      115 153 159 189 192 200 201 219 220 241 244 246 247 258 259 267 269—271
Representation      118
Rotation group      161 185 230
Rotation group, full      185
Schroedinger algebra      75 122 238
Schroedinger equation      73 121 145 157
Schroedinger equation, free particle      74 121 125 237 242
Schroedinger equation, harmonic oscillator      81 125 130 158
Schroedinger equation, linear potential      82 125 130 158
Schroedinger equation, radial free particle      106 109 112 113
Schroedinger equation, radial harmonic oscillator      106 108
Schroedinger equation, radial repulsive oscillator      106 109 111
Schroedinger equation, repulsive oscillator      82 87 125 130 158
Schroedinger equation, time dependent      73 158
Schroedinger equation, time-independent      157
Schroedinger group      77 86 122
Schroedinger group, covering group of      86
Self-adjoint operator      26 171
Semisplit coordinates      230 241
Separable coordinates      9—11 13—19 163 259
Separable coordinates for complex Helmholtz equation      61
Separable coordinates for Klein — Gordon equation      47
Separable coordinates for Laplace equation      207
Separable coordinates for Laplace operator on sphere      184
Separable coordinates for three-space Helmholtz equation      164
Separable coordinates for time-independent Schroedinger equation      157
Separable coordinates for two-space Helmholtz equation      22
Separable solutions      9—11 209
Separation constant      9
Separation of variables      9 11 39
Skew — Hermitian operator      24
Solid spherical harmonic      214 216—218
Solution space      1 171 204 206
Space inversion      60 185 205
Special conformal transformation      204 205 223 225
Special linear group      75 84 93 106 123 146 213 229 236 239 240
Special linear group, complex      97 113 115 222 247 250
Special linear group, universal covering of      85 108 240
Special orthogonal group, SO (4)      242
Special orthogonal group, SO(2, 1)      228 236 243
Special orthogonal group, SO(3)      161 172 174 184 185 228 230
Special orthogonal group, SO(3, 1)      242
Special orthogonal group, SO(3, 2)      228 230 233
Special orthogonal group, SO(4, 1)      205
Special orthogonal group, SO(4, 2)      242
Spectral decomposition      27 83 131 133 172 187 237
Sphere $S_{2}$       170 184
Spherical coordinates      106 165 170 172 176 191 192 198 201 203 207 212 214 218 219 222 233
Spherical harmonic      173 186 188 216 232
Spherical wave      176 195 202
Spheroidal wave equation      166
Spheroidal wave function      166
Split coordinates      230
Subgroup coordinates      13
Symmetric operator      25
Symmetry algebra      3 39 74 113 145 160 184 204 223 250 258
Symmetry group      2 205
Symmetry operator      2
Symmetry operator, first order      2—4 6 125 147 157
Symmetry operator, second order      6—8 20 40 60 125 147 157 162 184 206 213 230 239
Symmetry operator, trivial      7 162
Time translation      129
Toroidal coordinates      212 217
Transformation formula      155 203 222 241 253 267 269
Transformation group      4 39 161 205 224 250 262
Unitary representation      49
Unitary transformation      23 25 171 187
Vandermonde's theorem      255
Virial theorem      231
Wave equation      222 223 241 250
Weisner's method      62 98 105 113 118 152 155 198 202 213 218 254 259
Weyl group      76 92 122
Weyl's theorem      108
Whittaker function      181
Whittaker — Hill equation      135 141 167 181 218
Wigner D function      173

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