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arXiv · 1411.1024

Accurate approximations for the complex error function with small imaginary argument

S. M. Abrarov·B. M. Quine
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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.

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S. M. Abrarov, B. M. Quine. 2014-11-03. Accurate approximations for the complex error function with small imaginary argument. https://doi.org/10.5539/jmr.v1n1p44

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