# Algorithms for regular and irregular coulomb and bessel functions

## Abstract

Algorithms for computing Coulomb-Bessel functions are considered, with emphasis on obtaining accurate values when the argument x is inside the classical turning point xλ. Algorithms of Barnett et al. for the generalized Coulomb functions and their derivatives are discussed in the context of the phase integral formalism. Modified or alternative algorithms are considered that are designed to be valid for all values of argument x and index λ for the functions Fλ(x), Gλ(x). An algorithm for a ccelerating convergence of a power series by conversion to a continued fraction is presented, and is applied to the evaluation of spherical Bessel functions. An explicit formula for the integrand of the phase integral is presented for spherical Bessel functions. The methods considered need to be augmented by an efficient algorithm for computing the logarithmic derivative of G0 + iF0 for Coulomb functions when x is smaller than the charge parameter η. © 1984.