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Date: 25-3-2019
1203
Date: 23-5-2019
1598
Date: 28-8-2018
1358
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The wave equation in prolate spheroidal coordinates is
(1) |
where
(2) |
Substitute in a trial solution
(3) |
(4) |
The radial differential equation is
(5) |
and the angular differential equation is
(6) |
Note that these are identical (except for a sign change). The prolate angular function of the first kind is given by
(7) |
where is an associated Legendre polynomial. The prolate angular function of the second kind is given by
(8) |
where is an associated Legendre function of the second kind and the coefficients satisfy the recurrence relation
(9) |
with
(10) |
|||
(11) |
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(12) |
Various normalization schemes are used for the s (Abramowitz and Stegun 1972, p. 758). Meixner and Schäfke (1954) use
(13) |
Stratton et al. (1956) use
(14) |
Flammer (1957) uses
(15) |
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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أول صور ثلاثية الأبعاد للغدة الزعترية البشرية
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