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

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 $1\le |z|\le R$. 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.

arXiv comment24 pages, 2 figures

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Cited by in corpus (1)

• Solving Fredholm second-kind integral equations with singular right-hand sides on non-smooth boundaries