Arscott F.M. — Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions :: Электронная библиотека попечительского совета мехмата МГУ

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Arscott F.M. — Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions

Arscott F.M. — Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions

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Название: Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions

Автор: Arscott F.M.

Аннотация:

Of recent years, the two subjects of special functions and ordinary linear differential equations have (in this country at least) lain somewhat in the penumbra of a partial eclipse. Many mathematicians, not without reason, have come to regard the former as little more than a haphazard collection of ugly and unmemorable formulae, while attention in the latter has been directed mostly to existence-theorems and similar results for equations of general type. Only rarely does one find mention,at post-graduate level, of any problems in connection with the process of actually solving such equations. The electronic computer may perhaps be partly to blame for this, since the impression prevails in many quarters that almost any differential equation problem can be merely "put on the machine", so that finding an analytic solutionis largely a waste of time. This, however, is only a small part of the truth, for at the higher levels there are generally so many parameters or boundary conditions involved that numerical solutions, even if practicable, give no real idea of the properties of the equation. Moreover, any analyst of sensibility will feel that to fall back on numerical techniques savours somewhat of breaking a door with a hammer when one could, with a little trouble, find the key.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1964

Количество страниц: 281

Добавлена в каталог: 28.03.2010

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Предметный указатель

$ce_{n}$ -, $Ce_{n}$ -functions      see "Mathieu functions of integral order"
$ce_{\nu}$ -, $Ce_{\nu}$ -functions      see "Mathieu functions of fractional order"
$E^{m}_{n}$ -function      203—204 see
$F^{m}_{n}$ -functions      see "Lame functions of the second kind"
$Iso-\beta$ and $iso-\mu$ curves      128—129
$Mc^{(j)}$ , $Ms^{(j)}$ -functions      92 137
$me_{n}$ , $Me_{n}$ -functions      90—92 135—137
$me_{\nu}$ , $Me_{\nu}$ -functions      131—132 134—137 139—140
$M^{(j)}_{\nu}$ - functions      136—137
$se_{n}$ -, $Se_{n}$ -functions      see "Mathieu functions of integral order"
$se_{\nu}$ -, $Se_{\nu}$ -functions      see "Mathieu functions of fractional order"
$S^{\mu(j)}_{\nu}$ -functions      174—176 187—190 see
$\tilde{Q}$ -, $\tilde{Q}s$ -functions      186—187
Abel identity      28
Addition formulae for Mathieu functions      106
Associated Legendre equation      6 25 164—165
Associated Mathieu equation      2 14 27 153 182—185
Associated Mathieu functions      182—185
Asymptotic expansions for characteristic values      112—118
Asymptotic expansions for general Mathieu functions      139
Asymptotic expansions for Ince polynomials      151
Asymptotic expansions for Lame characteristic values      236
Asymptotic expansions for Mathieu functions      88 Chap.
Asymptotic expansions for spheroidal wave functions      186
Basically-periodic functions      31
Basically-periodic functions, solutions of Mathieu's equation      31 33—47 see
Bessel function product series for general Mathieu functions      138
Bessel function product series for Mathieu functions      98—99
Bessel function series for Associated Mathieu functions      185
Bessel function series for ellipsoidal wave functions      246—248
Bessel function series for Mathieu functions      86—88 89 135—136 138
Bessel function series for spheroidal wave functions      172—176 178
Bessel's equation, in wave problems      4 6
Bouwkamp      177 182
Brillouin      129 143
Burgess      72
Campbell      69 70 130—131 132 134 142 145 178 181 182 185
ceh-function      58
cel-, cdel-functions      see "Ellipsoidal wave functions"
cer-, cei-functions      104
ceu-function      137
Characteristic equation for Mathieu's equation      43 62 64
Characteristic exponent of Hill's equation      142—143 149
Characteristic exponent of Mathieu's equation      49 50 131
Characteristic exponent of spheroidal wave equation      161—165 169
Characteristic values for ellipsoidal wave functions      239
Characteristic values for Mathieu functions      10 43—46 62—70
Coexistence problem for Hill's equation      144
Coexistence problem for Mathieu's equation      34—37
Continued fractions, use in constructing Mathieu functions      61—64 67
del-function      see "Ellipsoidal wave functions"
Doubly-periodic equations      51
Dougall      79 99 100
Ec-, Es-functions      203 227 231
Eigenvalues      see "Characteristic values"
el-functions      see "Ellipsoidal wave functions"
Ellipsoidal coordinates      16—19 20 228—230
Ellipsoidal harmonics      25 214—218 228—230
Ellipsoidal wave equation      2 19 25 Chap.
Ellipsoidal wave functions of second and third kinds      248—250
Ellipsoidal wave functions, Bessel function series for      246—248
Ellipsoidal wave functions, Fourier — Jacobi series for      250
Ellipsoidal wave functions, integral relations      241—243
Ellipsoidal wave functions, main properties      238—241
Ellipsoidal wave functions, orthogonality      241
Ellipsoidal wave functions, perturbation series for      243—247
Ellipsoidal wave functions, power series for      250
Ellipsoidal wave functions, transformation formulae      250
Ellipsoidal wave functions, zeros      243
Elliptic coordinates      8 10 11 24
Elliptic membrane      10 150
Elliptic wave      84 104
Erdelyi      107 133 194 203 204 227—228
fe-, Fe-functions      see "Mathieu functions of integral order"
fek-, Fek-functions      see "Mathieu functions of the third kind"
fey-, Fey-functions      see "Mathieu functions of the second kind"
fl-function      249
Flammer      182
Floquet's theorem      29—31 48 194
Fourier — Jacobi series      220—222 227—228 250
ge-, ge-functions      see "Mathieu functions of integral order"
gek-, gey-functions      see "Mathieu functions of the third kind"
gey-, gey-functions      see "Mathieu functions of the second kind"
Goldstein      53—54 59 64 101 239
Halphen      213
Haupt      151
Heine      59
Heine, notation for Lame polynomials      202—204
hermite      16 194
Heun-type equations      231
Hill's equation      1 2 15—16 27 Chap. 190
Hill's equation with three terms (Whittaker — Hill equation)      2 16 23 43 141 144—146 150—151 238
Hill's method for determining stability      124—126 142—143
hl-function      249
Horn — Jeffreys technique      111 114—118
Humbert      145
Ince      2 34 47 74 101 104 144—145 182 185 194 203 227—228 231 232
Ince polynomials      148 149—152
Ince's equation (Whittaker — Hill equation)      145—152
Ince's theorem      34—37 65 77 125
Infinite determinant form of characteristic equation      64 67
Integral relations for ellipsoidal wave functions      241—243
Integral relations for Ince functions      150
Integral relations for Lame functions      211—217
Integral relations for Mathieu functions      40—42 79—84 85—86
Integral relations for spheroidal wave functions      42 180—181 184 188
Kelvin      59
Khabaza      231 236
Klotter and Kotowski      144
Lame functions (transcendental)      194 225—228
Lame functions (transcendental) of half-integral order      232
Lame functions (transcendental) of the second kind      223—225 230
Lame functions (transcendental) with single period      225—228 232
Lame polynomials      Chap. IX 239 246—248
Lame polynomials in potential theory      228—230
Lame polynomials, asymptotic formulae      236
Lame polynomials, expansion as series of      222—223
Lame polynomials, integral relations for      211—214
Lame polynomials, Legendre function series for      218—220
Lame polynomials, normalization      208—209
Lame polynomials, orthogonality      206—208
Lame polynomials, perturbation series for      231

Lame polynomials, tables      230—231
Lame polynomials, Tchebycheff polynomial series for      218—222
Lame polynomials, transformation of      209—211
Lame polynomials, zeros of      197—200 202—204 210—211 227 231 232—233
Lamp's equation      2 9 24—25 Chap. 237—238
Lamp's equation, algebraic forms      192—193
Lamp's equation, trigonometric form      193
Laplace's equation      14 19 191 213 228—230
Latent roots of matrices      20—22
Leitner and Spence      182
Lindemann — Stieltjes method for Mathieu's equation      49
Magnus      143—145 152
Malurkar      242
Markovic's proof of Ince's theorem      34—37
Mathieu      67
Mathieu function series      73—74 78 84—85 103
Mathieu functions of fractional order      133—134
Mathieu functions of integral order      51
Mathieu functions of second kind      37—38 70—72 76—77 89—96 98
Mathieu functions of third kind      91—98 136—137
Mathieu functions, addition formulae      106
Mathieu functions, Bessel function product series for      76 98—101
Mathieu functions, Bessel function series for      75—76 86—88 89—90
Mathieu functions, elementary relations      55—57
Mathieu functions, integral equations for      41—42 79—84 86 96—98
Mathieu functions, integral relations for      40 85—86 96—98
Mathieu functions, normalization      53—54 64 67 71
Mathieu functions, notation      52
Mathieu functions, orthogonality      57—58
Mathieu functions, perturbation method for      67—70 76—77
Mathieu functions, relations between different orders      103—104
Mathieu functions, tables      101
Mathieu functions, trigonometric series for      54—55 59—67
Mathieu functions, zeros of      47 111—114 120
Mathieu's equation      2 9 14 15 167—168
Mathieu's equation, algebraic form      11 47—49 72—73 74 78
Mathieu's equation, general theory      Chap. II
Mathieu's general equation      Chap. VI
Mathieu's general equation, notation for solutions      130—134
Mathieu's general equation, stable solutions      see "Mathieu functions of fractional order"
Mathieu's general equation, unstable solutions      134—135
Matrices (notation for)      21
McLachlan      9 54 64—65 82—83 129 132—134 137
Meissner equation      151
Meixner      44 69—70 74 92 106 119 128 136—137 169—170 176 179 181
Modified Mathieu equation      24 26 59
Modified Mathieu functions      58—59
Modified Mathieu functions, asymptotic behaviour      88
Moeglich      239 241 248
National Physical Laboratory      231
Ne-functions      92—98 137
Orthogonality of ellipsoidal wave functions      241
Orthogonality of Ince functions      149 151
Orthogonality of Lame polynomials      206—208
Orthogonality of Mathieu functions      39—40 50
Orthogonality of spheroidal wave functions      179 187 189
Oscillation      22
Paraboloidal coordinates      23
Periodic function (definition)      7
Periodicity equation      31—33 49 121
Periodicity exponent      31—33 49 121
Periodicity factor      31—33 49
Perron's rule      65—66 73 226—227
Perturbation method for ellipsoidal wave functions      243—246
Perturbation method for Ince functions      149—150
Perturbation method for Mathieu functions      67—69 138
Poincare      128—129
Ps-, Ps-functions      169—170 188—190 see
Pseudo-periodic solutions of doubly-periodic equations      51
Pseudo-periodic solutions of Mathieu's equation      31—34 49 127 138
Pseudo-periodic solutions of systems of equations      50
Qs-, Qs-functions      169—172 171—175 see
Roper      231
Schaefke      44 69—70 74 92 106 128 131 136—137 169—170
seh-function      58
sel-, scel-, sdel-, scdel-functions      see "Ellipsoidal wave functions"
Separation constant      3
Separation method of      1—2 6 9 13 18 23
seu-function      137
Shenitzer      143
Singularities of differential equations      1
Spherical Bessel functions      173
Spherical harmonics      25
Spherical harmonics, relations with ellipsoidal harmonics      214—218
Sphero-conal coordinates      24—25
Spheroidal coordinates      12 15 153 189
Spheroidal wave equation      2 12 27 Chap. 238
Spheroidal wave functions      159 163 176—181 186
Spheroidal wave functions, connexion with Associated Mathieu functions      182—183
Stability diagram for Mathieu's equation      122—124
Stability of solutions of periodic equations      50
Stable solutions of Mathieu's equation      121—122
Stable zone for Mathieu's equation      120
Stationary phase, method of      120
Stieltjes's theorem on Lame polynomials      197—198 233—234
Stokes phenomenon      110
Stratton et al.      181—182
Strutt      144
Sturm oscillation theorem      22—23
Sturm — Liouville expansions      74
Systems of periodic differential equations      50
Titchmarsh      74
Transcendental Ince functions      149
Transcendental Lame functions      194 226
Transformation of wave equation      13
uel-functions      see "Ellipsoidal wave functions"
Unstable solutions of Mathieu's equation      121—122
Urwin      151
Von Koppenfels      83
Wave equation in 2-dimensions      2 7
Wave equation in 3-dimensions      6 7 237—238
Wave equation in elliptic coordinates      8—10 59
Whittaker      53 145
Whittaker — Hill equation      16 23 43 142 144—146 150 238
Whittaker's $\sigma$ -method for Hill's equation      143
Whittaker's $\sigma$ -method for Mathieu functions      129 137
Winkler      143—145 152
Zeros of ellipsoidal wave functions      243
Zeros of Ince polynomials      148 149 150
Zeros of Lame functions      197—200 202—204 210—211 227—228 231 233
Zeros of Mathieu functions      10 47 54 74 111—114 120
Zeros of spheroidal wave functions      186
Zeros, Sturm's theorem on      22—23

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