Suggested Integration Technique for $\text{sinc}^2(x)/\sqrt{x^2+c^2}$? I'm looking for suggestions on how to approach the following integral:
$$\int_{-\infty}^{\infty} \frac{\text{sinc}^2(x)}{\sqrt{x^2 +c^2}}dx$$
where $c \in\ \mathbb{R}$ and $\text{sinc}(x):=\sin(x)/x$.
I've tried the residue theorem, but this failed because of branch points at $x=\pm ic$.
Am I missing an obvious approach, or is it likely that an analytic solution to this integral does not exist? Alternatively, are there any approaches for a clever analytic approximation to the integral? Any suggestions are greatly appreciated!
 A: I doubt that there is an explicit analytical solution to this problem. You can write the integral as
$$I=\mathcal{P} \int_{-\infty}^{\infty} \!dx\,\frac{1-e^{2 i x}}{2 x^2 \sqrt{c^2+x^2}} $$
You can deform the integration contour such that you only pick up the branch point at $x= ic$ (with the branch cut of the root along the imaginary axis). In this way, you can show that (I assume $c>0$)
$$I = \frac{\pi}{c} - 2\int_c^\infty \!d\xi\, \frac{1-e^{-2\xi}}{2 \xi^2 \sqrt{\xi^2 -c^2}}= \frac{\pi}{c} - \frac{1}{ c^2} + \int_c^\infty \!d\xi\, \frac{e^{-2\xi}}{\xi^2 \sqrt{\xi^2 -c^2}}$$ where $x=i \xi$.
We can get a good approximation for the remaining integral in the regime $c\gtrsim 1$ by expanding everything except for the exponential around $\xi =c$. We obtain the (asymptotic) expansion
$$\int_c^\infty \!d\xi\, \frac{e^{-2\xi}}{2 \xi^2 \sqrt{\xi^2 -c^2}} =
\int_c^\infty \!d\xi\, e^{-2\xi}\left(\frac{1}{\sqrt{2} c^{5/2} (\xi-c)^{1/2}} - \frac{9 (\xi-c)^{1/2}}{4 \sqrt{2}c^{7/2}} + O(\xi-c)^{3/2} \right) = \sqrt{\pi}e^{-2c} \left[\frac12c^{-5/2} -\frac{9}{32}c^{-7/2} +O(c^{-9/2}) \right]. $$
Thus, we have
$$ I = \frac{\pi}{c} -\frac{1}{c^2}$$
up to terms which decay as $e^{-2c}$. In particular,
$$ I = \frac{\pi}{c} -\frac{1}{c^2}+\sqrt{\pi}e^{-2c} \left[\frac12c^{-5/2} -\frac{9}{32}c^{-7/2} +O(c^{-9/2}) \right]$$
