Integral of sinc function multiplied by Gaussian I am wondering whether the following integral
$$\int_{-\infty}^{\infty}  \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$
exists in closed form. I would like to use it for numerical
calculation and find an efficient way to evaluate it.
If analytical form does not exist, I really appreciate
any alternative means for evaluating the integral.
One method would be numerical quadrature including
Gaussian quadrature, but it may be inefficient when
the parameters $a$ and $b$ are very different in scale.

EDIT : In view of this discussion, we have decided to add OP's self-answer to the end of the question, for it does not qualify as an answer yet contains vital details. The copying is unabridged.
Thanks very much for your comments, and the following result was obtained including the case for $x_0 \ne 0$:
$$
\int_{-\infty}^{\infty} dx \exp[-a(x-x_0)^2] \frac{ \sin(bx) }{ x }
= \pi \exp(-a x_0^2) \mathrm{Re}\left(\mathrm{erf}\left[\frac{b+2iax_0}{2\sqrt{a}}\right] - \mathrm{erf}\left[\frac{2iax_0}{2\sqrt{a}}\right]\right)
$$
where $a\gt0, b, x_0$ are assumed to be all real.
(note: coefficients etc may be still wrong...)
This integral appears in a type of electronic structure calculation
based on a grid representation (sinc-function basis). I believe
the above result should be definitely useful.
Thanks much!!
--jaian

We assume $a,b>0$. 
Then 
$$\begin{eqnarray*}
\int_{-\infty}^\infty dx\, e^{-a x^2}\frac{\sin b x}{x} 
&=& \int_0^b d\beta \, \int_{-\infty}^\infty dx\, e^{-a x^2} \cos \beta x \\
&=& \int_0^b d\beta \, \mathrm{Re} \int_{-\infty}^\infty dx\, e^{-a x^2+i \beta x} \\
&=& \int_0^b d\beta \, \mathrm{Re}\, \sqrt{\frac{\pi}{a}} e^{-{\beta}^2/(4a)} \\
&=& \pi \, \mathrm{erf}\left(\frac{b}{2\sqrt{a}}\right). 
\end{eqnarray*}$$
This approach can be generalized to $x_0\ne0$. 
