I am trying to find the following integral $$ I_\alpha(k,a,d) = \int_{-\infty}^\infty dx\, x^\alpha\frac{\exp(-ax^2+ixd)}{k^2-x^2+i0^+}, $$ where $\alpha$ is a nonzero integer, $k,d,a\in\mathbb R$, and $a>0$. Mathematica can do this integral for $d=0$,
$$ I_0(k,a,0) = \frac{\pi e^{-ak^2}[-i+\mathrm{erfi}(\sqrt{a}|k|)]}{|k|}, $$
and it is clear that $I_\alpha(k,0)$ is zero if $\alpha$ is odd and can otherwise be obtained from derivatives of the above expression.
Now for finite $d$, I'm at loss. I've tried to do a series expansion for small $d$, but it doesn't seem to converge nicely. But I feel like there should be a way to make some headway given the above result. Any pointers?
EDIT: We can also do the integral for $a=0$, but finite $d$ using contour integration. This yields $$ I_\alpha(k,0,d) = -i\pi e^{i|k d|}|k|^{\alpha-1}(\mathrm{sign}(d)) ^\alpha. $$
