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

## TL;DR

This paper presents a numerical scheme for accurately evaluating Abramowitz functions in the right half of the complex plane, combining series, asymptotic, and Laurent expansions for high precision.

## Contribution

It introduces a new computational method that efficiently evaluates Abramowitz functions using a combination of series, asymptotic, and Laurent expansions with precomputed coefficients.

## Key findings

- Achieves nearly machine precision for n=-1 to 2
- Uses a combination of series and asymptotic expansions
- Cost is about four times a complex exponential evaluation

## Abstract

A numerical scheme is developed for the evaluation of Abramowitz functions $J_n$ in the right half of the complex plane. For $n=-1,\, \ldots,\, 2$, 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 $1\le |z| \le R$. For $n>2$, $J_n$ is evaluated via a recurrence relation. The scheme achieves nearly machine precision for $n=-1, \ldots, 2$, with the cost about four times of evaluating a complex exponential per function evaluation.