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

Source abstract

Alternative expressions for calculating the prolate spheroidal radial functions of the second kind R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) and their first derivatives with respect to ξ \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 m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}(c, \xi ) and the prolate spheroidal angular function of the first kind S m l ( 1 ) ( c , η ) 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 m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) involves an integral of the product of S m l ( 1 ) ( c , η ) S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right ) and a spherical Neumann function kernel. It provides accurate values when ξ \xi is near unity and l − m l - m is not too large, even when c c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.