A sampling-based approximation of the complex error function and its implementation without poles

TL;DR

This paper introduces a sampling-based rational approximation method for the complex error function that avoids poles, enabling efficient and accurate computation across the entire complex plane, with a practical Matlab implementation.

Contribution

It develops a pole-free, sampling-based rational approximation of the complex error function for improved computational efficiency and accuracy.

Findings

Achieves high-accuracy approximation across the complex plane.

Provides an efficient Matlab implementation with only three approximations.

Avoids poles in the computation of the complex error function.

Abstract

Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function $w\left(z \right) = e^{- {z^2}}\left(1 + \frac{2i}{\sqrt \pi}\int_0^z e^{t^2}dt\right),$ where $z = x + iy$. As a further development, in this work we show how this sampling-based rational approximation can be transformed into alternative form for efficient computation of the complex error function $w\left(z \right)$ at smaller values of the imaginary argument $y=\operatorname{Im}\left[z \right]$. Such an approach enables us to avoid poles in implementation and to cover the entire complex plain with high accuracy in a rapid algorithm. An optimized Matlab code utilizing only three rapid approximations is presented.