  In our recent publications we have introduced the incomplete cosine expansion
of the sinc function for efficient application in sampling [Abrarov & Quine,
Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7
(2) (2015) 163-174]. Here we show that it can also be utilized as a flexible
and efficient tool in mathematical analysis. In particular, an application of
the incomplete cosine expansion of the sinc function leads to expansion series
of the error function in form of a sum of the Gaussian functions. This approach
in integration provides a new asymptotic formula for the constant $\pi$.
