A generalized series expansion of the arctangent function based on the enhanced midpoint integration

TL;DR

This paper introduces a generalized series expansion for the arctangent function using enhanced midpoint integration, significantly improving computational accuracy and efficiency, especially for high-precision calculations of pi.

Contribution

It presents a novel generalized series expansion derived from EMI that enhances accuracy and convergence rate without complex numbers or surds.

Findings

Improved accuracy in arctangent computation with increasing subintervals

Rapid convergence rate in calculating digits of pi

Potential for practical high-precision applications

Abstract

In this work we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes two-step iteration without surd and complex numbers. The computational test we performed reveals that such a generalization improves accuracy in computation of the arctangent function by many orders of the magnitude with increasing integer $M$, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of $\pi$ in the Machin-like formulas.