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Spheroidal Wave Function

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Stratton (1935), Chu and Stratton (1941), and Rhodes (1970) define the spheroidal functions as those solutions of the differential equation

(1-eta^2)psi_(alphan)^('')(c,eta)-2(alpha+1)etapsi_(alphan)^'(c,eta)+(b_(alphan)-c^2eta^2)psi_(alphan)(c,eta)=0
(1)

that remain finite at the singular points eta=+/-1. The condition of finiteness restricts the admissible values of the parameter b_(alphan)(c) to a discrete set of eigenvalues indexed by n=0, 1, 2, ... (Rhodes 1970).

The radial solution R_(mn)(xi) in prolate spheroidal coordinates satisfies the differential equation

d/(dxi)[(xi^2-1)d/(dxi)R_(mn)(xi)] -(lambda_(mn)-c^2xi^2+(m^2)/(xi^2-1))R_(mn)(xi)=0
(2)

and the angular solution S_(mn)(eta) satisfies

d/(deta)[(1-eta^2)d/(deta)S_(mn)(eta)] +(lambda_(mn)-c^2eta^2-(m^2)/(1-eta^2))S_(mn)(eta)=0.
(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 gamma=npi/2 and m=1.

The angular wave functions have series expansions about gamma=0 given by

PS_(nm)=P_n^m(x)+[((-m+n+1)(-m+n+2)P_(n+2)^m(x))/(2(2n+1)(2n+3)^2) -((m+n-1)(m+n)P_(n-2)^m(x))/(2(2n-1)^2(2n+1))]gamma^2+O(gamma^3) QS_(nm)=Q_n^m(x)+[((-m+n+1)(-m+n+2)Q_(n+2)^m(x))/(2(2n+1)(2n+3)^2) -((m+n-1)(m+n)Q_(n-2)^m(x))/(2(2n-1)^2(2n+1))]gamma^2+O(gamma^3).
(4)

The radial wavefunctions have asymptotic behavior as z->infty given by

S_(nm)^((1))(z)∼1/(gammaz)sin(gammaz-1/2npi)
(5)
S_(nm)^((2))(z)∼1/(gammaz)cos(gammaz-1/2npi).
(6)

Whittaker and Watson (1990, p. 403) call

S_(mn)^((1))=2pi((n-m)!)/((n+m)!)P_n^m(ir)P_n^m(costheta)cos; sin(mphi)
(7)
S_(mn)^((2))=2pi((n-m)!)/((n+m)!)Q_n^m(ir)Q_n^m(costheta)cos; sin(mphi),
(8)

where P_l^m(x) is a Legendre polynomial and Q_l^m(x) 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

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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.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.

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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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