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Zbl 1066.65029
van Buren, Arnie L.; Boisvert, Jeffrey E.
Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. (English)
[J] Q. Appl. Math. 62, No. 3, 493-507 (2004). ISSN 0033-569X; ISSN 1552-4485/e

Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the second kind $R^{(2)}_{ml}(c,\xi)$ 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.\par The first alternative expression is obtained from the expansion of the product of $R^{(2)}_{ml}(c,\xi)$ 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. 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.\par The second alternative expression for $R^{(2)}_{ml}(c,\xi)$ involves an integral of the product of $S^{(1)}_{ml}(c,\eta)$ 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.

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 of the second kind; prolate spheroidal angular function of the first kind

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