Rapid Computation of the Plasma Dispersion Function: Rational and Multi-pole Approximation, and Improved Accuracy

TL;DR

This paper presents optimized rational and multi-pole approximations for the plasma dispersion function Z(s), achieving high accuracy with twelve significant digits and providing a valuable computational resource for plasma physics applications.

Contribution

It introduces optimized coefficients and methods for rapid, accurate computation of Z(s), improving accuracy especially in the intermediate argument range.

Findings

Achieves twelve-digit precision in computing Z(s)

Provides optimized coefficients for rational and multi-pole approximations

Enhances accuracy across the entire argument range

Abstract

The plasma dispersion function $Z(s)$ is a fundamental complex special integral function widely used in the field of plasma physics. The simplest and most rapid, yet accurate, approach to calculating it is through rational or equivalent multi-pole expansions. In this work, we summarize the numerical coefficients that are practically useful to the community. Besides the Pade approximation to obtain coefficients, which are accurate for both small and large arguments, we also employ optimization methods to enhance the accuracy of the approximation for the intermediate range. The best coefficients provided here for calculating $Z(s)$ can deliver twelve significant decimal digits. This work serves as a foundational database for the community for further applications.