TITLE: A tricky integral
QUESTION [6 upvotes]: Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral
$$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$
I tried several changes of variables, and contour integration doesn't seem to work. 
Motivation comes from the following closely related kernel
$$K(s,t)=e^{-\frac{(s-t)^2}{4}}\int_{-\infty}^{\infty}(1+x^2)^{-1/2}e^{-\frac{(s-t)^2}{4} x^2}e^{-i (s^2-t^2) x}dx,$$ 
which provides an example of a compact integral operator on $L^2(\mathbb{R})$ that is not Hilbert-Schmidt. I would like to check the details. 
Thank you!

REPLY [0 votes]: You can express it as a (formal) sum over terms which involve the Hermite polynomials, in the form $HermiteH(n,\frac{\beta}{2\sqrt{\alpha}})$.  You can compute the exact term from
$$\int_{-\infty}^{\infty} x^n e^{-\alpha x^2}e^{-i\beta x}dx.$$
Since this involves summing over the index of a polynomial, the techniques I know stop working.
Note that for $\alpha=0$ this is $-\frac{\pi}{2}\left( Y_0(\beta i) + Y_0(-\beta i)\right)$ (again, Bessel function).  So with Neil's answer for $\beta=0$ and the 'hint' that this integral satisfies the Heat equation (see Zen's comment), you get another equivalent formulation of the problem.
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