A rational approximation of the arctangent function and a new approach\n in computing pi

2016/03/08 by Sanjar M. Abrarov, Abrarov, S. M., Brendan M. Quine +3
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Diverse Scientific and Engineering Research #FOS: Mathematics #General Mathematics (math.GM) #Mathematical functions and polynomials #Scientific Research and Discoveries

paper · pdf · doi:10.48550/arxiv.1603.03310

openalex publication_date 2016/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We have shown recently that integration of the error function\n rmerf\( x \) represented in form of a sum of the Gaussian\nfunctions provides an asymptotic expansion series for the constant pi. In this\nwork we derive a rational approximation of the arctangent function arctan\n\( x \) that can be readily generalized it to its counterpart -\n rmsgn\( x \)\π /2 + arctan \( x \), where\n rmsgn\( x \) is the signum function. The application of the\nexpansion series for these two functions leads to a new asymptotic formula for\n\π.\n

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