I have problems with the following integral:
$$\int_{-\infty}^{\infty}e^{-ax^2+ibx^2}\left(\frac{1}{c/i+x}+\frac{1}{c/i-x}\right)dx,$$
where $a,b$ and $c$ are real numbers.
This screams Cauchy's integral formula to me, but i can't find a contour that works.
Note that if the pole is in the positive (negative) complex half-plane, the $i\sin$-part escapes in the negative (positive) complex half plane.
Any help would be much appreciated.