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

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 $π= 4 \arctan \left( 1 \right)$. 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 $π$ can be computed with high accuracy.