“Prolate Spheroidal Wave Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Prolate Spheroidal Wave Function

释义

Prolate Spheroidal Wave Function

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)
			(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)

See also Oblate Spheroidal Wave Function

References

Abramowitz, M. and Stegun, C. 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.

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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