On the accurate computation of the prolate spheroidal radial functions of the second kind

The series expansion of the prolate radial functions of the second kind, expressed in terms of the spherical Neumann functions, converges very slowly when the spheroid’s surface coordinate $\xi$ approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of $\left ( {{\xi ^2} - 1} \right )$ is obtained to facilitate the convergence. Then, by using the Wronskian test, it is shown that this newly developed expansion has been computed with a double precision accuracy.