Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives
Authors:
Arnie L. Van Buren and Jeffrey E. Boisvert
Journal:
Quart. Appl. Math. 62 (2004), 493-507
MSC:
Primary 33F05; Secondary 33E10, 65D20
DOI:
https://doi.org/10.1090/qam/2086042
MathSciNet review:
MR2086042
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Abstract: Alternative expressions for calculating the prolate spheroidal radial functions of the second kind $R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right )$ and their first derivatives with respect to $\xi$ are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of $R_{ml}^{\left ( 2 \right )}(c, \xi )$ 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. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all parameter ranges. The second alternative expression for $R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right )$ involves an integral of the product of $S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )$ and a spherical Neumann function kernel. It provides accurate values when $\xi$ is near unity and $l - m$ is not too large, even when $c$ becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.
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- Arnie L. Van Buren and Jeffrey E. Boisvert, Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives, Quart. Appl. Math. 60 (2002), no. 3, 589–599. MR 1914443, DOI https://doi.org/10.1090/qam/1914443
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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)
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 and J. E. Boisvert, Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives, Quart. Appl. Math. 60, 589–599 (2002)
J. Meixner and F. W. Schäfke, Mathieusche Funckionen und Sphäroidfunckionen, Springer-Verlag, Berlin, 1954
C. Flammer, Spheroidal Wave Functions, Stanford Univ. Press, Stanford, Calif., 1957
C. J. Bouwkamp, Theoretical and numerical treatment of diffraction through a circular aperture, IEEE Trans. Antennas and Propagation AP-18, 152–176 (1970)
M. Abramowitz and C. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, Dover, New York, NY, 1972
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© Copyright 2004
American Mathematical Society