Exactification of the asymptotics for Bessel and Hankel functions

摘要

Exactification is the process of showing how a complete asymptotic expansion can be evaluated to yield exact values of the original function it represents. Because the process does not involve neglecting the remainders of truncated component asymptotic series within a complete asymptotic expansion, techniques for evaluating divergent series are required. One such technique is Borel summation, but in many instances, it can be computationally slow and consequently, lacking in precision. Another is the numerical technique of Mellin–Barnes regularization, which is capable of evaluating divergent series with great precision far more rapidly than Borel summation. Here, both techniques are presented in the evaluation of exact values for Bessel and Hankel functions from their complete asymptotic expansions.