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

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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 ξ \xi approaches 1 (thin spheroids). In this paper an analytical series expansion in powers of ( ξ 2 − 1 ) \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.