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

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 $y=\mathrm{Im}\left[z\right]$. 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.

20 pages