  We have shown recently that integration of the error function
${\rm{erf}}\left( x \right)$ 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 $ -
{\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
$\pi$.
