Evaluation of Abramowitz functions in the right half of the complex plane

Zydrunas Gimbutas; Shidong Jiang; Li-Shi Luo

doi:10.1016/j.jcp.2019.109169

paper

Evaluation of Abramowitz functions in the right half of the complex plane

arXiv:1906.12023 · doi:10.1016/j.jcp.2019.109169

Abstract

A numerical scheme is developed for the evaluation of Abramowitz functions $J_{n}$ in the right half of the complex plane. For $n = - 1, \dots, 2$ , the scheme utilizes series expansions for $∣ z ∣ < 1$ and asymptotic expansions for $∣ z ∣ > R$ with $R$ determined by the required precision, and modified Laurent series expansions which are precomputed via a least squares procedure to approximate $J_{n}$ accurately and efficiently on each sub-region in the intermediate region $1 \leq ∣ z ∣ \leq R$ . For $n > 2$ , $J_{n}$ is evaluated via a recurrence relation. The scheme achieves nearly machine precision for $n = - 1, \dots, 2$ , with the cost about four times of evaluating a complex exponential per function evaluation.

24 pages, 2 figures