Oblate 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:
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22-7-2019
1682
Oblate Spheroidal Wave Function
The wave equation in oblate 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-x_2^2))(partial^2Phi)/(partialphi^2)
+c^2(xi_1^2+xi_2^2)Phi=0,](http://mathworld.wolfram.com/images/equations/OblateSpheroidalWaveFunction/NumberedEquation1.gif) |
(1)
|
where
 |
(2)
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Substitute in a trial solution
 |
(3)
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The radial 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))R_(mn)(c,xi_2)=0,](http://mathworld.wolfram.com/images/equations/OblateSpheroidalWaveFunction/NumberedEquation4.gif) |
(4)
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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))R_(mn)(c,xi_2)=0](http://mathworld.wolfram.com/images/equations/OblateSpheroidalWaveFunction/NumberedEquation5.gif) |
(5)
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(Abramowitz and Stegun 1972, pp. 753-755; Zwillinger 1997, p. 127).
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.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997.
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