Sampling by incomplete cosine expansion of the sinc function: application to the Voigt/complex error function

TL;DR

This paper introduces a novel sampling method using incomplete cosine expansion for the sinc function, leading to highly accurate and efficient rational approximations of the complex error function, beneficial for fast computations.

Contribution

It presents a new sampling technique based on incomplete cosine expansion, improving accuracy and efficiency over traditional sinc-based methods for the complex error function.

Findings

Achieves higher accuracy than Weideman's approximation with fewer terms.

Enables integration using only elementary functions.

Demonstrates efficiency and practicality through numerical tests.

Abstract

A new sampling methodology based on incomplete cosine expansion series is presented as an alternative to the traditional sinc function approach. Numerical integration shows that this methodology is efficient and practical. Applying the incomplete cosine expansion we obtain a rational approximation of the complex error function that with the same number of the summation terms provides an accuracy exceeding the Weideman\text{'}s approximation accuracy by several orders of the magnitude. Application of the expansion results in an integration consisting of elementary function terms only. Consequently, this approach can be advantageous for accurate and rapid computation.