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MacRobert T.M. — Spherical Harmonics an Elementary Treatise on Harmonic Functions with Applications
MacRobert T.M. — Spherical Harmonics an Elementary Treatise on Harmonic Functions with Applications

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Название: Spherical Harmonics an Elementary Treatise on Harmonic Functions with Applications

Автор: MacRobert T.M.

Язык: en

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 1947

Количество страниц: 372

Добавлена в каталог: 13.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Addition theorems      see “Bessel coefficients and functions” “Zonal
Analytical continuation      304 307
Associated Legendre functions of the first kind      122 314—345
Associated Legendre functions of the first kind, asymptotic expansions of      322 323
Associated Legendre functions of the first kind, definite integral expressions for      124 323
Associated Legendre functions of the first kind, Dougall's formulae for      326—328
Associated Legendre functions of the first kind, expansion in terms of      336—339
Associated Legendre functions of the first kind, Ferrer's      125 126 314—343
Associated Legendre functions of the first kind, integrals of products of      126 142
Associated Legendre functions of the first kind, Laplace's integrals for      127 129
Associated Legendre functions of the first kind, Mehlor — Dirichlat integral for      323
Associated Legendre functions of the first kind, recurrence formulae for      124 142 143 329—331
Associated Legendre functions of the first kind, relations between      124 323
Associated Legendre functions of the first kind, scries for      123 315
Associated Legendre functions of the second kind      122 316—345
Associated Legendre functions of the second kind, asymptotic expansions of      322
Associated Legendre functions of the second kind, definite integral expressions for      124
Associated Legendre functions of the second kind, recurrence formulae for      124 142 143
Associated Legendre functions of the second kind, relations between      125 316
Associated Legendre functions of the second kind, scries for      123 316 318 365
Asymptotic expansions      see “Bessel functions” “Ilypergeometric
Attraction, gravitational      144
Attraction, of a plane lamina      152
Axes of spherical harmonics      234
Axes of zonal harmonics      131 137
Bessel coefficients      250
Bessel coefficients, addition theorems for      252
Bessel coefficients, expansions in terms of      251
Bessel functions      241 312 313 342—344
Bessel functions, addition theorems for      252 260 271
Bessel functions, applications of      286—297
Bessel functions, asymptotic expansions of      272
Bessel functions, definite integral expressions for      255 265 271 272 342—344
Bessel functions, expansion for ${x}^{n}$ in terms of      253
Bessel functions, integrals involving      275 285 302 304 311 343. 367
Bessel functions, modified      269 342—344
Bessel functions, of the first kind      244
Bessel functions, of the second kind, ${G}_{n}$      248 257
Bessel functions, of the second kind, ${Y}_{n}$      246 248
Bessel functions, products of      364
Bessel functions, recurrence formulae for      245 259 270
Bessel functions, relations between      259 271
Bessel functions, whose order is half an odd integer      250 285
Bessel functions, zeros of      277—283
Bessel's equation      241
Bessel's equation, solution of      242 255
Bessel's integral      251
Beta junction      83
Binomial expansion, remainder in      80
boundary conditions      29
Boundary problems      168 170 228
Calory      24
Christoffel's summation formulae      100 115 335
Clerk Maxwell's theory of spherical harmonics      231
Coaxal circles      219
Coefficients, Laplace's      see “Laplace's coefficients”
Coefficients, Legendre      see “Legendre coefficients”
Conduction of heat      23
Conduction of Heat, equation of      27 30
Conduction of heat, in circular cylinder      255 294 295 296
Conduction of heat, in finiterod      33 35 39 41 49 50
Conduction of heat, in Fourier's ring      36
Conduction of heat, in infinite rectangular solid      41
Conduction of heat, in infinite rod      46 50
Conduction of heat, in infinite slab      31
Conduction of heat, in infinite solid      48 49
Conduction of heat, in rectangular parallelepiped      43 45
Conduction of heat, in semi-infinite solid      48
Conduction of heat, in sphere      292 290
Conduction of heat, in wedge      295
Conduction of heat, with radiation at surface      34
Conductivity, thermal      23
Conductivity, thermometric      28
Conductor, distribution of electricity on      179—194 203 222—226
Convection of heat      23 29
Convergence      313
Cross-cut      307
Curvilinear co-ordinates      195
Cylinder functions      242
Cylinder, flow of heat in      288 294 295 296
Cylindrical harmonics      242
Diffusivity      28
Dipolar co-ordinates      219 221
Dipolar co-ordinates, Laplace's equation in terms of      226
Dipolar co-ordinates, reciprocal of distance in terms of      222
Dirichlet's conditions for a Fourier expansion      6
Dirichlet's integrals      7
Discontinuity, point of      7 11
Domain      299
Duhamel's theorem      38
Eccentric spheres      191 222 297
Electrostatics      179
Ellipsoid of revolution, oblate      194 198 212
Ellipsoid of revolution, prolate      198 202 203 209 210
Elliptic co-ordinates      199 213
Elliptic co-ordinates, reciprocal of distance in terms of      204 216
Emissivity      29
Equivalent distributions of masses      167
Euler's constant      82
for Legendre functions      98 112 113 124 142 143 329—332
Fourier series      1
Fourier series, convergence of      11 12
Fourier series, differentiation of      12
Fourier series, sine and cosine      12
Fourier series, summation of      10
Fourier — Bessel expansions      283
Fourier's double integral      16 48 49
Fourier's expansion      1 13
Fourier's expansion, at a point of discontinuity      11
Fourier's expansion, Dirichlet's conditions for      6
Fourier's expansion, for odd and even functions      3 4
Fourier's expansion, validity of      5
Fourier's ring      36
Gamma function      82 340 349
Gauss's equation      77 298
Gauss's theorem for the hypergeometric function      83 300
Gauss's theorem on attractions      146
Generalised hypergeometric functions      310 313
Green      72
Green's function      186
Green's theorem      27
Harmonics      59
Harmonics, cylindrical      242
Harmonics, sectorial      132
Harmonics, spherical      see “Spherical harmonics”
Harmonics, tesseral      132
Harmonics, zonal      131 137
Harp string, vibrations of      60
Heat      see “Conduction” “Convection” “Radiation”
Heat, specific      24
Hypergeometric equation      77 298
Hypergeometric equation, relations between the integrals of      305
Hypergeometric function      77 295—313
Hypergeometric function, asymptotic expansion of      80 309
Hypergeometric function, four forms of      79 303
Hypergeometric function, Gauss's theorem for      83
Hypergeometric series      77 313
Indicial equation      78 109 243
Initial conditions      29
Inverse points      161
Isothermal surfaces      24
Jacobi's proof of Rodrigues' formula      89
Kelvin      73
Kelvin's theorem      74
Krammer's function      311
Lamp's functions      218
Laplace      72 73
Laplace's coefficients      73 75 136
Laplace's coefficients, addition theorem for      137
Laplace's equation      74 147
Laplace's equation, transformation of      74 195 199 213 226 230 241 297
Laplace's integrals      101 102 127 129
legendre      72
Legendre coefficients      72 76 85 131
Legendre coefficients, applications to potential theory of      144
Legendre coefficients, as derivatives      105 238
Legendre coefficients, axes of      131
Legendre coefficients, ChristofTel's summation formula, for      100 115
Legendre coefficients, expansions of      86 87 88 89 104 106
Legendre coefficients, expansions of functions in scries of      100 102 338 339
Legendre coefficients, expression for potential in terms of      148 150
Legendre coefficients, expression for solid angle in terms of      153
Legendre coefficients, expressions for polynomials in terms of      95 97 106
Legendre coefficients, integrals involving      91 93 94 96 104 105 106 107 117
Legendre coefficients, Laplace's integrals for      101 102
Legendre coefficients, poles of      131 238
Legendre coefficients, potential of circular lamina in terms of      152
Legendre coefficients, potential of magnet in terms of      155
Legendre coefficients, potential of magnetic shell in terms of      156
Legendre coefficients, recurrence formulae for      98
Legendre coefficients, Rodrigue's formula for      88
Legendre coefficients, summation formulae for      100 115
Legendre coefficients, zeros of      91 131
Legendre factions of the second kind      108 116 117 118
Legendre factions of the second kind, addition theorem for      355
Legendre factions of the second kind, recurrence formula for      113
Legendre factions of the second kind, relations connecting with Legendre functions of the first kind      114 115 116 117 118
Legendre functions of the first kind      111 119
Legendre functions of the first kind, addition theorem for      328
Legendre functions of the first kind, integrals involving      117
Legendre functions of the first kind, recurrence formulae for      112
Legendre functions, associated      “Associated Legendre functions”
Legendre's associated equation      76 122 314 316
Legendre's associated equation, relations between the integrals of      316—321
Legendre's equation      76 90
Legendre's equation, solution of      108 in
Legendre's expansion      72 85
Lommel integrals      275
Loops of vibrating strings      59
Magnet      155
Magnetic shell      156
Membranes, vibrations of,      see “Vibrations of membranes”
modes, fundamental      58
Modes, normal, of vibrating strings      56
Neighbourhood of a point      6
Neumann's addition formula for ${J}_{0}$      252
Newton's Law of attraction      144
Nodal lines      71
Nodes of vibrating strings      59
overtones      59
Piano string, vibrations of      65
Point souuv of heat      69
Poisson's equation      145 195
Poisson's integrals      103 169
Poles oi spheiieal harmonies      234 238
Poles oi zonal harmonies      131 137 23s
Polynomials, Legendre      see “Legendre coefficients”
Potential      72 145 146 157 158
Potential, derivative of, in terms of spherical harmonics      150
Potential, in terms ol spherical harmonics      148
Potential, of circular lamina      154
Potential, of circular wire      151
Potential, of concentric spheres      169
Potential, of eccentric spheres      191 222
Potential, of ellipsoid of revolution      203 209 210
Potential, of hemispherical shell      152
Potential, of magnet      155
Potential, of magnetic shell      156
Potential, of singular point      235
Potential, of sphere      166 180—188
Potential, of spherical shell      158—167 165 185—191
Potential, of spheroid      172 175 191—194
Potential, of spheroidal shell      177
Radiation of heat      23
Radiation of heat, at surface      29 34 37 39 41
Recurrence formulae, for Bebscl functions      248 . 270
Relations between Bessel and Legendre functions      339—345 363 366
Remainder in binomial expansion      80
Ring functions      230
Ring surfaces      225
Ring, Fourier's      36
Rodrigues' formula      88
Rodrigues' formula, Jacobi's proof of      89
Rodrigues' formula, the extended      334
Rodrigues' general formula      89
Sectorial harmonics      132 238
Singular points      235
Source of heat, point      49
Specific heat      24
Spheres concentric      169
Spheres, boundary problems for      168 170
Spheres, conduction of heat in      292. 290
Spheres, eccentric      191 222 297
Spheres, potentials of      160 169 180—188
Spherical harmonics      73
Spherical harmonics, axes of      234
Spherical harmonics, Clerk Maxwell's theory of      231
Spherical harmonics, expansions in terms of      138 141
Spherical harmonics, expression of potential in terms of      148
Spherical harmonics, expressions of      236
Spherical harmonics, in terms of zonal, tesseral, and sectorial harmonics      133
Spherical harmonics, integrals involving      135 137
Spherical harmonics, poles of      234
Spherical harmonics, potential of shells and spheies in terms of      158 166
Spherical harmonics, solid      74 77
Spherical harmonics, surface      73 75 77 131 132 133 135 236
Spherical shells, potentials of      157—168 188—191
Spheroidal shells, potentials of      177
Spheroids, potentials of      172—178 191—194
Stirling's formula      83
Strings, vibrations of      see “Vibrations of strings”
Surface conditions      29
Tait      73
Tesseral harmonics      132 238
thermal conductivity      23
Thermometric conductivity      28
Thomson      73
Tone, fundamental      59
Vibrations of strings      52
Vibrations of strings, differential equation of      52 55
Vibrations of strings, energy of      54 60
Vibrations of strings, for semi-infinite string      55
Vibrations of strings, for string fixed at ends      56
Vibrations of strings, for unlimited string      55
Vibrations of strings, forced      54 66
Vibrations of strings, free      54
Vibrations of strings, fundamental tones of      59
Vibrations of strings, harmonics of      59
Vibrations of strings, loops of      59
Vibrations of strings, nodes of      59
Vibrations of strings, normal modes of      56
Vibrations of strings, of harp      60
Vibrations of strings, of piano      65
Vibrations of strings, of violin      63
Vibrations of strings, overtones of      59
Vibrations, circular      286 364
Vibrations, of membranes      69
Vibrations, rectangular      70
Violin string, vibrations of      63
Waves      55 56 61
Wedge, conduction of heat in      295 296
Whitehead, R.F.      340
Zeros of Legendre coefficients      91 131
Zeros, of Bessel functions      277 283
Zonal harmonics      131
Zonal harmonics, addition theorem for      137
Zonal harmonics, axes of      131. 137
Zonal harmonics, general      137
Zonal harmonics, poles of      131 137 238
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