A new asymptotic expansion series for the constant pi

TL;DR

The paper introduces a novel asymptotic expansion series for pi derived from the incomplete cosine expansion of the sinc function, offering an efficient analytical tool for mathematical analysis and error function expansion.

Contribution

It presents a new asymptotic formula for pi based on the incomplete cosine expansion, expanding its application beyond sampling to mathematical analysis.

Findings

Derivation of a new asymptotic series for pi

Representation of error functions as sums of Gaussian functions

Enhanced methods for mathematical analysis involving pi

Abstract

In our recent publications we have introduced the incomplete cosine expansion of the sinc function for efficient application in sampling [Abrarov & Quine, Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7 (2) (2015) 163-174]. Here we show that it can also be utilized as a flexible and efficient tool in mathematical analysis. In particular, an application of the incomplete cosine expansion of the sinc function leads to expansion series of the error function in form of a sum of the Gaussian functions. This approach in integration provides a new asymptotic formula for the constant $\pi$.