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

We have shown recently that integration of the error function $\mathrm{e}\mathrm{r}\mathrm{f}\left(x\right)$ 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 $\arctan \left( x \right)$thatcanbereadilygeneralizedittoitscounterpart$ - {\rm{sgn}}\left( x \right)\pi /2 + \arctan \left( x \right)$,where${\rm{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 $\pi$.