cv.complex variables - Contour integration problem from probability

Now, since you call erfc(1) "a closed form expression", I should confess I do not understand the rules of this game. What's the big difference between $int_1^infty e^{-x^2/2} dx$ and the original integral? Or, do you ask if it is an elementary function of the parameter $c$?

If the latter, note that the function $J(c)=e^{c^2}int_{-infty}^inftyfrac{e^{-(x-c)^2}}{1+x^2}dx$ satisfies the equation $J''+4J=4sqrtpi e^{c^2}$, which, if you try to solve it by the method of variation of parameters, leads to the indefinite integrals like $int e^{c^2}cos 2c dc$. Those are not elementary, but not much worse than your erfc.

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