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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. (English) Zbl 1031.65034
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. {\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.

65D20Computation of special functions, construction of tables
33E10Lamé, Mathieu, and spheroidal wave functions
33F05Numerical approximation and evaluation of special functions