A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

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 a two-step iteration without surd or complex numbers. The computational test we performed reveal...

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Bibliographic Details
Published in:AppliedMath 2023-06, Vol.3 (2), p.395-405
Main Authors: Abrarov, Sanjar M., Siddiqui, Rehan, Jagpal, Rajinder Kumar, Quine, Brendan M.
Format: Article
Language:English
Subjects:
arctangent function
constant pi
iterative algorithm
midpoint integration
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Summary: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 a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of 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 π in the Machin-like formulas.
ISSN:2673-9909
2673-9909
DOI:10.3390/appliedmath3020020