Stratton (1935), Chu and Stratton (1941), and Rhodes (1970) define the spheroidal functions as those solutions of the differential equation
(1)
that remain finite at the singular points . The condition of finiteness restricts the admissible
values of the parameter to a discrete set of eigenvalues indexed by ,
1, 2, ... (Rhodes 1970).
The radial solution in prolate spheroidal coordinates satisfies the differential
equation
(2)
and the angular solution satisfies
(3)
Note that the differential equations are identical, so the radial and angular wavefunctions satisfy the same differential equation over different ranges of the variable (Abramowitz and Stegun 1972, p. 753).
Angular spheroidal harmonics are implemented in the Wolfram Language as SpheroidalPS [n ,
m , gamma , x ] and SpheroidalQS [n ,
m , gamma , x ]; radial spheroidal harmonics are implemented as
SpheroidalS1 [n ,
m , gamma , x ] and SpheroidalS2 [n ,
m , gamma , x ]; and eigenvalues are implemented as SpheroidalEigenvalue [n ,
m , gamma ].
Spheroidal wave functions become elementary if and .
The angular wave functions have series expansions about given by
(4)
The radial wavefunctions have asymptotic behavior as given by
Whittaker and Watson (1990, p. 403) call
where
is a Legendre polynomial and is a Legendre
function of the second kind the internal and external spheroidal wavefunctions.
However, they are not true spheroidal wave functions in the usual sense of the word.
See also Ellipsoidal Harmonic of the First Kind ,
Ellipsoidal
Harmonic of the Second Kind ,
Oblate
Spheroidal Wave Function ,
Prolate
Spheroidal Wave Function ,
Spherical Harmonic
Explore with Wolfram|Alpha
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. Chu, L. J. and Stratton,
J. A. "Elliptic and Spheroidal Wave Functions." J. Math. and Phys. 20 ,
259-309, 1941. Falloon, P. "Homepage of the Spheroidal Wave Functions."
http://www.physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html . Falloon,
P. E.; Abbott, P. C.; and Wang, J. B. "Theory and Computation
of the Spheroidal Wave Functions." 18 Dec 02. http://arxiv.org/abs/math-ph/0212051 . Falloon,
P. E. Theory and Computation of Spheroidal Harmonics with General Arguments.
Masters thesis. Perth, Australia: University of Western Australia, 2001. http://www.physics.uwa.edu.au/pub/Theses/2002/Falloon/Masters_Thesis.pdf . Flammer,
C. Spheroidal
Wave Functions. Stanford, CA: Stanford University Press, 1957. Meixner,
J. and Schäfke, R. W. Mathieusche Funktionen und Sphäroidfunktionen.
Berlin: Springer-Verlag, 1954. Morse, P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 642-644,
1953. Rhodes, D. R. "On the Spheroidal Functions." J.
Res. Nat. Bur. Standards--B. Math. Sci. 74B , 187-209, Jul.-Sep. 1970. Stratton,
J. A. "Spheroidal Functions." Proc. Nat. Acad. Sci. 21 ,
51-56, 1935. Stratton, J. A.; Morse, P. M; Chu, L. J.;
and Hutner, R. A. Elliptic
Cylinder and Spheroidal Wave Functions, including Tables of Separation Constants
and Coefficients. New York: Wiley, 1941. Stratton, J. A.;
Morse, P. M.; Chu, L. J.; Little, J. D. C.; and Corbató,
F. J. Spheroidal
Wave Functions. New York: Wiley, 1956. Whittaker, E. T.
and Watson, G. N. A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, 1990. Referenced on Wolfram|Alpha Spheroidal Wave Function
Cite this as:
Weisstein, Eric W. "Spheroidal Wave Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SpheroidalWaveFunction.html
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