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Zbl 1031.65034
Van Buren, Arnie L.; Boisvert, Jeffrey E.
Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. (English)
[J] Q. Appl. Math. 60, No.3, 589-599 (2002). ISSN 0033-569X; ISSN 1552-4485/e

Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind $R^{(1)}_{ml}(c,\xi)$ 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 $R^{(1)}_{ml}(c,\xi)$ and the prolate spheroidal angular function of the first kind $S^{(1)}_{ml}(c,\eta)$ in a series of products of the corresponding spherical functions.\par {\it B. J. King} and {\it A. L. van Buren} [SIAM J. Math. Anal. 4, 149-160 (1973; Zbl 0249.33011)] 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 $R^{(1)}_{ml}(c,\xi)$ and its first derivative.

MSC 2000:: *65D20 Computation of special functions
33E10 Spheroidal wave functions, etc.
33F05 Numerical approximation of special functions

Keywords: Helmholtz wave equation; prolate spheroidal radial functions; first derivatives; prolate spheroidal angular function; convergence

Citations: Zbl 0249.33011

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