Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi
For numerical analysts, this offers a flexible method for high-precision pi computation, but it is an incremental improvement over existing integration techniques.
The paper presents an enhanced midpoint integration method combined with Taylor expansion to derive arctangent identities for high-accuracy computation of pi. With small L and M values, pi is computed with high accuracy.
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.