Asymptotic Matching the Self-Consistent Expansion to Approximate the Modified Bessel Functions of the Second Kind

TL;DR

This paper introduces a method combining the self-consistent expansion with asymptotic matching to derive uniform approximations for modified Bessel functions of the second kind, improving the accuracy of perturbative solutions.

Contribution

The paper proposes a novel approach to use asymptotic matching with the self-consistent expansion for uniform approximation of special functions.

Findings

Successfully applied to approximate modified Bessel functions of the second kind.

Provides a systematic way to tune approximations to match asymptotic behaviour.

Eliminates the need for empirical comparison to validate approximations.

Abstract

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem - too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, $K_\alpha(x)$.