Prolate Spheroidal Wave Function
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
23-7-2019
2024
Prolate Spheroidal Wave Function
The wave equation in prolate spheroidal coordinates is
![del ^2Phi+k^2Phi=partial/(partialxi_1)[(xi_1^2-1)(partialPhi)/(partialxi_1)]+partial/(partialxi_2)[(1-xi_2^2)(partialPhi)/(partialxi_2)]
+(xi_1^2-xi_2^2)/((xi_1^2-1)(1-xi_2^2))(partial^2Phi)/(partialphi^2)+c^2(xi_1^2-xi_2^2)Phi=0,](http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation1.gif) |
(1)
|
where
 |
(2)
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Substitute in a trial solution
 |
(3)
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![d/(dxi_1)[(xi_1^2-1)d/(dxi_1)R_(mn)(c,xi_1)]-(lambda_(mn)-c^2xi_1^2+(m^2)/(xi_1^2-1))R_(mn)(c,xi_1)=0.](http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation4.gif) |
(4)
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The radial differential equation is
![d/(dxi_2)[(xi_2^2-1)d/(dxi_2)S_(mn)(c,xi_2)]-(lambda_(mn)-c^2xi_2^2+(m^2)/(xi_2^2-1))R_(mn)(c,xi_2)=0,](http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation5.gif) |
(5)
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and the angular differential equation is
![d/(dxi_2)[(1-xi_2^2)d/(dxi_2)S_(mn)(c,xi_2)]-(lambda_(mn)-c^2xi_2^2+(m^2)/(1-xi_2^2))S_(mn)(c,xi_2)=0.](http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation6.gif) |
(6)
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Note that these are identical (except for a sign change). The prolate angular function of the first kind is given by
![S_(mn)^((1))=<span style=]() {sum_(r=1,3,...)^inftyd_r(c)P_(m+r)^m(eta) for n-m odd; sum_(r=0,2,...)^inftyd_r(c)P_(m+r)^m(eta) for n-m even, " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation7.gif" style="height:64px; width:285px" /> |
(7)
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where 
is an associated Legendre polynomial. The prolate angular function of the second kind is given by
![S_(mn)^((2))=<span style=]() {sum_(r=...,-1,1,3,...)d_r(c)Q_(m+r)^m(eta) for n-m odd; sum_(r=...,-2,0,2,...)d_r(c)Q_(m+r)^m(eta) for n-m even, " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation8.gif" style="height:86px; width:293px" /> |
(8)
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where 
is an associated Legendre function of the second kind and the coefficients 
satisfy the recurrence relation
 |
(9)
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with
Various normalization schemes are used for the 
s (Abramowitz and Stegun 1972, p. 758). Meixner and Schäfke (1954) use
![int_(-1)^1[S_(mn)(c,eta)]^2deta=2/(2n+1)((n+m)!)/((n-m)!).](http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation10.gif) |
(13)
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Stratton et al. (1956) use
![((n+m)!)/((n-m)!)=<span style=]() {sum_(r=1,3,...)^(infty)((r+2m)!)/(r!)d_r for n-m odd; sum_(r=0,2,...)^(infty)((r+2m)!)/(r!)d_r for n-m even. " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation11.gif" style="height:102px; width:293px" /> |
(14)
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Flammer (1957) uses
![S_(mn)(c,0)=<span style=]() {P_n^(m+1)(0) for n-m odd; P_n^m(0) for n-m even. " src="http://mathworld.wolfram.com/images/equations/ProlateSpheroidalWaveFunction/NumberedEquation12.gif" style="height:46px; width:230px" /> |
(15)
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REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Spheroidal Wave Functions." Ch. 21 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 751-759, 1972.
Flammer, C. Spheroidal Wave Functions. Stanford, CA: Stanford University Press, 1957.
Meixner, J. and Schäfke, F. W. Mathieusche Funktionen und Sphäroidfunktionen. Berlin: Springer-Verlag, 1954.
Rhodes, D. R. "On the Spheroidal Functions." J. Res. Nat. Bur. Standards--B. Math. Sci. 74B, 187-209, Jul.-Sep. 1970.
Stratton, J. A.; Morse, P. M.; Chu, L. J.; Little, J. D. C.; and Corbató, F. J. Spheroidal Wave Functions. New York: Wiley, 1956.
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