A new application methodology of the Fourier transform for rational approximation of the complex error function

TL;DR

This paper introduces a novel Fourier transform-based method for rational approximation of the complex error function, achieving high accuracy with minimal computational effort suitable for spectroscopic applications.

Contribution

The paper proposes a new Fourier transform application methodology that yields an efficient rational approximation of the complex error function with high accuracy and computational speed.

Findings

Achieves average accuracy of 10^-15 with 17 summation terms

Provides rapid computation without trigonometric or exponential functions

Effective over a wide domain relevant for spectroscopy

Abstract

This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides the average accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.