Is there an analytic expression for the following integral? Is there a complex plane contour that would accomplish the task? $$\int_{-\infty}^\infty \frac{e^{-x^2}}{x-a}\,dx$$ where $\Im(a)\ne 0$
The contour can not take as its part a half circle going through either the upper or the lower part at infinity of the complex plane, since either path makes the exponential term explode.