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



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

јвтор: Arscott F.M.

јннотаци€:

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.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1964

 оличество страниц: 281

ƒобавлена в каталог: 28.03.2010

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный 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      203Ч204 see
$F^{m}_{n}$-functions      see "Lame functions of the second kind"
$Iso-\beta$ and $iso-\mu$ curves      128Ч129
$Mc^{(j)}$, $Ms^{(j)}$-functions      92 137
$me_{n}$, $Me_{n}$-functions      90Ч92 135Ч137
$me_{\nu}$, $Me_{\nu}$-functions      131Ч132 134Ч137 139Ч140
$M^{(j)}_{\nu}$- functions      136Ч137
$se_{n}$-, $Se_{n}$-functions      see "Mathieu functions of integral order"
$se_{\nu}$-, $Se_{\nu}$-functions      see "Mathieu functions of fractional order"
$S^{\mu(j)}_{\nu}$-functions      174Ч176 187Ч190 see
$\tilde{Q}$-, $\tilde{Q}s$-functions      186Ч187
Abel identity      28
Addition formulae for Mathieu functions      106
Associated Legendre equation      6 25 164Ч165
Associated Mathieu equation      2 14 27 153 182Ч185
Associated Mathieu functions      182Ч185
Asymptotic expansions for characteristic values      112Ч118
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 33Ч47 see
Bessel function product series for general Mathieu functions      138
Bessel function product series for Mathieu functions      98Ч99
Bessel function series for Associated Mathieu functions      185
Bessel function series for ellipsoidal wave functions      246Ч248
Bessel function series for Mathieu functions      86Ч88 89 135Ч136 138
Bessel function series for spheroidal wave functions      172Ч176 178
Bessel's equation, in wave problems      4 6
Bouwkamp      177 182
Brillouin      129 143
Burgess      72
Campbell      69 70 130Ч131 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      142Ч143 149
Characteristic exponent of Mathieu's equation      49 50 131
Characteristic exponent of spheroidal wave equation      161Ч165 169
Characteristic values for ellipsoidal wave functions      239
Characteristic values for Mathieu functions      10 43Ч46 62Ч70
Coexistence problem for Hill's equation      144
Coexistence problem for Mathieu's equation      34Ч37
Continued fractions, use in constructing Mathieu functions      61Ч64 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      16Ч19 20 228Ч230
Ellipsoidal harmonics      25 214Ч218 228Ч230
Ellipsoidal wave equation      2 19 25 Chap.
Ellipsoidal wave functions of second and third kinds      248Ч250
Ellipsoidal wave functions, Bessel function series for      246Ч248
Ellipsoidal wave functions, Fourier Ч Jacobi series for      250
Ellipsoidal wave functions, integral relations      241Ч243
Ellipsoidal wave functions, main properties      238Ч241
Ellipsoidal wave functions, orthogonality      241
Ellipsoidal wave functions, perturbation series for      243Ч247
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 227Ч228
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      29Ч31 48 194
Fourier Ч Jacobi series      220Ч222 227Ч228 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      53Ч54 59 64 101 239
Halphen      213
Haupt      151
Heine      59
Heine, notation for Lame polynomials      202Ч204
hermite      16 194
Heun-type equations      231
Hill's equation      1 2 15Ч16 27 Chap. 190
Hill's equation with three terms (Whittaker Ч Hill equation)      2 16 23 43 141 144Ч146 150Ч151 238
Hill's method for determining stability      124Ч126 142Ч143
hl-function      249
Horn Ч Jeffreys technique      111 114Ч118
Humbert      145
Ince      2 34 47 74 101 104 144Ч145 182 185 194 203 227Ч228 231 232
Ince polynomials      148 149Ч152
Ince's equation (Whittaker Ч Hill equation)      145Ч152
Ince's theorem      34Ч37 65 77 125
Infinite determinant form of characteristic equation      64 67
Integral relations for ellipsoidal wave functions      241Ч243
Integral relations for Ince functions      150
Integral relations for Lame functions      211Ч217
Integral relations for Mathieu functions      40Ч42 79Ч84 85Ч86
Integral relations for spheroidal wave functions      42 180Ч181 184 188
Kelvin      59
Khabaza      231 236
Klotter and Kotowski      144
Lame functions (transcendental)      194 225Ч228
Lame functions (transcendental) of half-integral order      232
Lame functions (transcendental) of the second kind      223Ч225 230
Lame functions (transcendental) with single period      225Ч228 232
Lame polynomials      Chap. IX 239 246Ч248
Lame polynomials in potential theory      228Ч230
Lame polynomials, asymptotic formulae      236
Lame polynomials, expansion as series of      222Ч223
Lame polynomials, integral relations for      211Ч214
Lame polynomials, Legendre function series for      218Ч220
Lame polynomials, normalization      208Ч209
Lame polynomials, orthogonality      206Ч208
Lame polynomials, perturbation series for      231
Lame polynomials, tables      230Ч231
Lame polynomials, Tchebycheff polynomial series for      218Ч222
Lame polynomials, transformation of      209Ч211
Lame polynomials, zeros of      197Ч200 202Ч204 210Ч211 227 231 232Ч233
Lamp's equation      2 9 24Ч25 Chap. 237Ч238
Lamp's equation, algebraic forms      192Ч193
Lamp's equation, trigonometric form      193
Laplace's equation      14 19 191 213 228Ч230
Latent roots of matrices      20Ч22
Leitner and Spence      182
Lindemann Ч Stieltjes method for Mathieu's equation      49
Magnus      143Ч145 152
Malurkar      242
Markovic's proof of Ince's theorem      34Ч37
Mathieu      67
Mathieu function series      73Ч74 78 84Ч85 103
Mathieu functions of fractional order      133Ч134
Mathieu functions of integral order      51
Mathieu functions of second kind      37Ч38 70Ч72 76Ч77 89Ч96 98
Mathieu functions of third kind      91Ч98 136Ч137
Mathieu functions, addition formulae      106
Mathieu functions, Bessel function product series for      76 98Ч101
Mathieu functions, Bessel function series for      75Ч76 86Ч88 89Ч90
Mathieu functions, elementary relations      55Ч57
Mathieu functions, integral equations for      41Ч42 79Ч84 86 96Ч98
Mathieu functions, integral relations for      40 85Ч86 96Ч98
Mathieu functions, normalization      53Ч54 64 67 71
Mathieu functions, notation      52
Mathieu functions, orthogonality      57Ч58
Mathieu functions, perturbation method for      67Ч70 76Ч77
Mathieu functions, relations between different orders      103Ч104
Mathieu functions, tables      101
Mathieu functions, trigonometric series for      54Ч55 59Ч67
Mathieu functions, zeros of      47 111Ч114 120
Mathieu's equation      2 9 14 15 167Ч168
Mathieu's equation, algebraic form      11 47Ч49 72Ч73 74 78
Mathieu's equation, general theory      Chap. II
Mathieu's general equation      Chap. VI
Mathieu's general equation, notation for solutions      130Ч134
Mathieu's general equation, stable solutions      see "Mathieu functions of fractional order"
Mathieu's general equation, unstable solutions      134Ч135
Matrices (notation for)      21
McLachlan      9 54 64Ч65 82Ч83 129 132Ч134 137
Meissner equation      151
Meixner      44 69Ч70 74 92 106 119 128 136Ч137 169Ч170 176 179 181
Modified Mathieu equation      24 26 59
Modified Mathieu functions      58Ч59
Modified Mathieu functions, asymptotic behaviour      88
Moeglich      239 241 248
National Physical Laboratory      231
Ne-functions      92Ч98 137
Orthogonality of ellipsoidal wave functions      241
Orthogonality of Ince functions      149 151
Orthogonality of Lame polynomials      206Ч208
Orthogonality of Mathieu functions      39Ч40 50
Orthogonality of spheroidal wave functions      179 187 189
Oscillation      22
Paraboloidal coordinates      23
Periodic function (definition)      7
Periodicity equation      31Ч33 49 121
Periodicity exponent      31Ч33 49 121
Periodicity factor      31Ч33 49
Perron's rule      65Ч66 73 226Ч227
Perturbation method for ellipsoidal wave functions      243Ч246
Perturbation method for Ince functions      149Ч150
Perturbation method for Mathieu functions      67Ч69 138
Poincare      128Ч129
Ps-, Ps-functions      169Ч170 188Ч190 see
Pseudo-periodic solutions of doubly-periodic equations      51
Pseudo-periodic solutions of Mathieu's equation      31Ч34 49 127 138
Pseudo-periodic solutions of systems of equations      50
Qs-, Qs-functions      169Ч172 171Ч175 see
Roper      231
Schaefke      44 69Ч70 74 92 106 128 131 136Ч137 169Ч170
seh-function      58
sel-, scel-, sdel-, scdel-functions      see "Ellipsoidal wave functions"
Separation constant      3
Separation method of      1Ч2 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      214Ч218
Sphero-conal coordinates      24Ч25
Spheroidal coordinates      12 15 153 189
Spheroidal wave equation      2 12 27 Chap. 238
Spheroidal wave functions      159 163 176Ч181 186
Spheroidal wave functions, connexion with Associated Mathieu functions      182Ч183
Stability diagram for Mathieu's equation      122Ч124
Stability of solutions of periodic equations      50
Stable solutions of Mathieu's equation      121Ч122
Stable zone for Mathieu's equation      120
Stationary phase, method of      120
Stieltjes's theorem on Lame polynomials      197Ч198 233Ч234
Stokes phenomenon      110
Stratton et al.      181Ч182
Strutt      144
Sturm oscillation theorem      22Ч23
Sturm Ч 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      121Ч122
Urwin      151
Von Koppenfels      83
Wave equation in 2-dimensions      2 7
Wave equation in 3-dimensions      6 7 237Ч238
Wave equation in elliptic coordinates      8Ч10 59
Whittaker      53 145
Whittaker Ч Hill equation      16 23 43 142 144Ч146 150 238
Whittaker's $\sigma$-method for Hill's equation      143
Whittaker's $\sigma$-method for Mathieu functions      129 137
Winkler      143Ч145 152
Zeros of ellipsoidal wave functions      243
Zeros of Ince polynomials      148 149 150
Zeros of Lame functions      197Ч200 202Ч204 210Ч211 227Ч228 231 233
Zeros of Mathieu functions      10 47 54 74 111Ч114 120
Zeros of spheroidal wave functions      186
Zeros, Sturm's theorem on      22Ч23
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