Each terms in this equation is analytically integrable. Therefore, its integra-
tion leads to the expansion series (3) of the error function. The more detailed
description of the expansion series (3) of the error function is given in our
work [2].
References
[1] H.A. Fayed and A.F. Atiya, An evaluation of the integral of the
product of the error function and the normal probability density
with application to the bivariate normal integral, Math. Comp., 83
(2014) 235-250. http://www.ams.org/journals/mcom/2014-83-285/
S0025-5718-2013-02720-2/
[2] S.M. Abrarov and B.M. Quine, A new asymptotic expansion series for
the constant pi, arXiv:1603.01462
[3] J.M. Borwein, P.B. Borwein and D.H. Bailey, Ramanujan, modular equa-
tions, and approximations to pi or how to compute one billion digits of pi,
Amer. Math. Monthly, 96 (3) (1989) 201-219. http://www.jstor.org/
stable/2325206
[4] J.M. Borwein and S.T. Chapman, I prefer pi: A brief history and an-
thology of articles in the American Mathematical Monthly, Amer. Math.
Monthly, 122 (3) (2015) 195-216. http://dx.doi.org/10.4169/amer.
math.monthly.122.03.195
[5] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2
nd
ed.,
Chapman & Hall/CRC 2003.
[6] E.W. Ng and M. Geller, A table of integrals of the error functions, J.
Research Natl. Bureau Stand. 73B (1) (1969) 1-20. http://dx.doi.org/
10.6028/jres.073B.001
[7] S.M. Abrarov and B.M. Quine, A rational approximation for efficient
computation of the Voigt function in quantitative spectroscopy, J. Math.
Research, 7 (2) (2015) 163-174. http://dx.doi.org/10.5539/jmr.v7n2
Preprint version: arXiv:1504.00322
[8] J.H. Mathews and K.D. Fink, Numerical methods using Matlab, 4
th
ed.,
Prentice Hall 1999.
10