Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives

Authors:
Arnie L. Van Buren and Jeffrey E. Boisvert

Journal:
Quart. Appl. Math. **60** (2002), 589-599

MSC:
Primary 65D20; Secondary 33E12

DOI:
https://doi.org/10.1090/qam/1914443

MathSciNet review:
MR1914443

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Abstract: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and their first derivatives with respect to $\xi$ are shown to provide accurate values, even for low values of $l - m$ where the traditional expressions provide increasingly inaccurate results as the size parameter $c$ increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and the prolate spheroidal angular function of the first kind $S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )$ in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and its first derivative.

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B. J. King and A. L. Van Buren, *A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives*, Naval Research Lab. Rpt. 7161 (1970)
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S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King, *Tables of radial spheroidal wave functions, volumes* 4, 5, 6, *oblate*, $m = 0, 1, 2$, Naval Research Lab. Rpts. 7091, 7092, and 7093 (1970)
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B. J. King, R. V. Baier, and S. Hanish, *A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives*, Naval Research Lab. Rpt. 7012 (1970)
B. J. King and A. L. Van Buren, *A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives*, Naval Research Lab. Rpt. 7161 (1970)
S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King, *Tables of radial spheroidal wave functions, volumes* 1, 2, 3, *prolate*, $m = 0, 1, 2$, Naval Research Lab. Rpts. 7088, 7089, and 7090 (1970)
A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish, *Tables of angular spheroidal wave functions, vol. I, prolate*, $m = 0$, Naval Research Lab. Publication, U.S. Govt. Printing Office (1975)
A. L. Van Buren, R. V. Baier, and S. Hanish, *A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives*, Naval Research Lab. Rpt. 6959 (1969)
S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King, *Tables of radial spheroidal wave functions, volumes* 4, 5, 6, *oblate*, $m = 0, 1, 2$, Naval Research Lab. Rpts. 7091, 7092, and 7093 (1970)
A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish, *Tables of angular spheroidal wave functions, vol. II, oblate*, $m = 0$, Naval Research Lab. Publication, U.S. Govt. Printing Office (1975)
B. J. Patz and A. L. Van Buren, *A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind*, Naval Research Lab. Memo. Rpt. 4414 (1981)
A. L. Van Buren, *A Fortran computer program for calculating the linear prolate functions*, Naval Research Lab. Rpt. 7994 (1976)
J. Meixner and F. W. Schäfke, *Mathieusche Funktionen und Sphäroidfunktionen, mit Anwendungen auf physikalische und technische Probleme*, Die Grundlehren der mathematischen Wissenschaften, Vol. 71, Springer-Verlag, Berlin, 1954
C. Flammer, *Spheroidal Wave Functions*, Stanford Univ. Press, Stanford, CA, 1957
B. J. King and A. L. Van Buren, *A general addition theorem for spheroidal wave functions*, SIAM J. Math. Anal. **4**, 149–160 (1973)
B. P. Sinha and R. H. Macphie, *Translational addition theorems for spheroidal scalar and vector wave functions*, Quart. Appl. Math. **38**, 143–158 (1980)
R. H. Macphie, J. Dalmas, and R. Deleuil, *Rotational-translational addition theorems for scalar spheroidal wave functions*, Quart. Appl. Math. **44**, 737–749 (1987)
C. J. Bouwkamp, *Theoretical and numerical treatment of diffraction through a circular aperture*, IEEE Trans. Antennas and Propagation **AP-18**, 152–176 (1970)
L. J. Chu and J. A. Stratton, *Elliptic and spheroidal wave functions*, J. Appl. Phys. **20**, 259–309 (1941)
P. M. Morse and H. Feshbach, *Methods of Theoretical Physics*, Parts I and II, McGraw-Hill, New York, 1953
L. Page, *The electrical oscillations of a prolate spheroid* II, III, Phys. Rev. **65**, 98–117 (1944)
E. Whittaker and G. Watson, *A Course in Modern Analysis*, 4th edition, Cambridge, 1952
M. Abramowitz and C. A. Stegun, *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, 9th printing, Dover, NY, 1972

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© Copyright 2002
American Mathematical Society