Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi

S. M. Abrarov; B. M. Quine

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arXiv · 1604.03752

Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi

S. M. Abrarov·B. M. Quine

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Abstract

We describe a method of integration to obtain identities of the arctangent function and show how this method can be applied to the high-accuracy computation of the constant pi using the equation $\pi = 4 \arctan \left( 1 \right)$. Our approach combines the midpoint method with the Taylor expansion series to enhance accuracy in the subintervals. The accuracy of this method of integration is determined by number of subintervals $L$ and by order of the Taylor expansion $M$. This approach provides significant flexibility in computation since the required convergence in resulting equations can be optimized through appropriate choices for the integers $L$ and $M$. Sample computations are presented to illustrate that even with relatively small values of the integers $L$ and $M$ the constant $\pi$ can be computed with high accuracy.

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BibTeX RIS

S. M. Abrarov, B. M. Quine. 2016-04-10. Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi. https://arxiv.org/abs/1604.03752

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