The higher derivatives of the inverse tangent function and rapidly convergent BBP-type formulas

TL;DR

This paper derives a closed-form expression for the derivatives of arctan x, introduces a new expansion for arctan x, and presents rapidly convergent series for calculating pi and pi times sqrt 3.

Contribution

It provides a novel closed formula for higher derivatives of arctan x and new rapidly convergent series for pi, enhancing computational methods.

Findings

Closed formula for the n-th derivative of arctan x

New expansion for arctan x

Rapidly convergent series for pi and pi*sqrt(3)

Abstract

We give a closed formula for the $n^{th}$ derivative of $\arctan x$. A new expansion for $\arctan x$ is also obtained and rapidly convergent series for $\pi$ and $\pi\sqrt 3$ are derived.