A rational approximation of the arctangent function and a new approach in computing pi

Abstract

We have shown recently that integration of the error function $\mathrm{erf}\left(x\right)$ represented in form of a sum of the Gaussian functions provides an asymptotic expansion series for the constant pi. In this work we derive a rational approximation of the arctangent function $\arctan \left(x\right)$ that can be readily generalized it to its counterpart $-\mathrm{sgn}\left(x\right)\pi /2+\arctan \left(x\right)$, where $\mathrm{sgn}\left(x\right)$ is the signum function. The application of the expansion series for these two functions leads to a new asymptotic formula for $\pi$.

10 pages, 2 figures