Efficient application of the Chiarella and Reichel series approximation of the complex error function

TL;DR

This paper presents a simplified derivation of a series approximation for the complex error function using Fourier expansion, and introduces an algorithm for its accelerated and accurate computation.

Contribution

It provides a new derivation method for the series approximation of the CEF and proposes an algorithm for its efficient and accurate evaluation.

Findings

Derived a simple Fourier-based series for the CEF

Identified a lower bound constraint for the input parameter y

Developed an algorithm for high-accuracy, accelerated CEF computation

Abstract

Using the theorem of residues Chiarella and Reichel derived a series that can be represented in terms of the complex error function (CEF). Here we show a simple derivation of this CEF series by Fourier expansion of the exponential function $\exp ({- {\tau ^2}/4})$. Such approach explains the existence of the lower bound for the input parameter $y = \operatorname{Im} [z]$ restricting the application of the CEF approximation. An algorithm resolving this problem for accelerated computation of the CEF with sustained high accuracy is proposed.