Accurate approximations for the complex error function with small imaginary argument

TL;DR

This paper introduces two efficient approximations for the complex error function with small imaginary parts, achieving high accuracy and computational speed, useful for applications requiring rapid and precise calculations.

Contribution

The paper presents novel approximations for the complex error function using Dawson's integral, optimized for small imaginary arguments and high accuracy over a specific domain.

Findings

First approximation exceeds 10^{-9} accuracy in real part

Second approximation exceeds 10^{-13} accuracy in real part

Second approximation offers high accuracy and is computationally efficient

Abstract

In this paper we present two efficient approximations for the complex error function $w \left( {z} \right)$ with small imaginary argument $\operatorname{Im}{\left[ { z } \right]} < < 1$ over the range $0 \le \operatorname{Re}{\left[ { z } \right]} \le 15$ that is commonly considered difficult for highly accurate and rapid computation. These approximations are expressed in terms of the Dawson's integral $F\left( x \right)$ of real argument $x$ that enables their efficient implementation in a rapid algorithm. The error analysis we performed using the random input numbers $x$ and $y$ reveals that in the real and imaginary parts the average accuracy of the first approximation exceeds ${10^{ - 9}}$ and ${10^{ - 14}}$, while the average accuracy of the second approximation exceeds ${10^{ - 13}}$ and ${10^{ - 14}}$, respectively. The first approximation is slightly faster in computation. However, the second approximation provides excellent high-accuracy coverage over the required domain.