Looking for guidance on a Fourier integral Working with a Fourier transform problem, I've encountered the following integral:
$$
\int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx
$$
where $a$, $b$, and $c$ are real coefficients.  Mathematica claims that this has no closed-form solution, but I suspect (hope?) that's not the case.  Unfortunately, my background in complex analysis is limited to some panic-studying I did to make it through a field theory course a decade ago.  I've spent the weekend dusting that off, but am still stumped...
I know there's a pole at $z=ci$.  If I take (to me) the obvious extension of the integral to the complex plane by simply replacing $x$ with $z$, I can even calculate its residue as:
$$
\frac{\exp\left(a^2c^2-bc\right)}{2ci}
$$
My hope was to integrate over a semicircle in the positive half-plane, and use this residue to get the integral along the real axis.  However, with this extension, the integral over the semicircle of radius R doesn't seem to be zero (or possibly even converge) with R going to infinity -- or at least, Jordan's lemma doesn't give me any reason to believe so, since the $-a^2z^2$ part becomes unfriendly on the imaginary axis.
I vaguely recall there being a strategy for dealing with this sort of problem.  I specifically recall the strategy not being to use $zz^*$, since explicit dependence on the complex conjugate was a no-go.  For the life of me, though, I can't recall what the strategy was.
If anybody can (a) refresh my memory on this; (b) give a solution method; or (c) explain why no closed form solution exists, that would be much appreciated.  Thanks in advance for any help you can offer!
 A: The gaussian term makes complex analysis iffy.  This integral may be done much  more easily using the convolution theorem.
To begin, note that
$$\int_{-\infty}^{\infty} dx \, e^{-a x^2} e^{i k x} = \sqrt{\frac{\pi}{a}} e^{-k^2/(4 a)}$$
$$\int_{-\infty}^{\infty} dx \frac{e^{i k x}}{x^2+c^2} = \frac{\pi}{c} e^{-c |k|}$$
The integral sought is the convolution of these transforms.  That is,
$$\int_{-\infty}^{\infty} dx \frac{e^{-a x^2}}{x^2+c^2} e^{i b x} = \frac{1}{2 \pi} \sqrt{\frac{\pi}{a}} \frac{\pi}{c} \int_{-\infty}^{\infty} dk \,e^{-k^2/(4 a)} \, e^{-c |b-k|} $$
This integral is very doable, despite what Mathematica says.  For the time being, I will spare you the details, which involve splitting the integral about $k=b$ and rescaling to get a nice form.  The result is that the sought-after integral is
$$\frac{\pi}{c} e^{a c^2} \left [e^{-b c} \left (1+\text{erf}\left (\frac{b-2 a c}{2 \sqrt{a}} \right ) \right ) + e^{b c} \left (1-\text{erf}\left (\frac{b+2 a c}{2 \sqrt{a}} \right ) \right )\right ]  $$
where erf is the error function.
NOTE
My $a$ is the OP's $a^2$ - this was not intentional, but to get an answer the OP wants, sub $a^2$ for $a$ in the above result.
