A difficult integral Is there any analytical result on the following integral?
$$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+e^{-(x-\mu)}}dx$$
Thanks a lot!
 A: The integral is a special case of what is called Gauss–Fermi function in semiconductor physics. In general, it cannot be calculated analytically. However some semi-analytical approximations do exist. See, for example, http://link.springer.com/article/10.1007/s10825-014-0615-7 (Analytical evaluation of charge carrier density of organic materials with Gauss density of states, by T. Mehmetoğlu) and references therein.
A: Maple can evaluate this integral for integer and half-integer values of $\mu$. The values seem have a nice pattern.  Let the value of the integral be $F(\mu)$.  Then $F(0)= \frac12\sqrt\pi$ and for integer $\mu$ with $1\le\mu\le 30$, 
$$ F(\mu) = - \sqrt\pi\,e^{-\mu^2} \biggl(\frac12 + \sum_{i=1}^{2\mu-1} (-1)^i e^{i^2/4}\Biggr).$$
There's also an obvious symmetry that gives $F(-\mu)+F(\mu)=\sqrt\pi$.
For half-integers, the empirical pattern is $F(\mu+\frac12)=F(-\mu)e^{-\mu-1/4}$ (checked for lots of integer $\mu$).
I assume these are all easy to prove but I'll leave that for someone else.
