Like the title says, I am trying to evaluate
$\int_{-\infty}^{\infty} dx \cdot x e^{iax}e^{-ic\sqrt{(x^2 + b^2)}}$
I'm not a mathematician by trade, but I ran across this integral in my research and am lost with how to evaluate it. $a,b,c$ are positive and real. We also assume $a$ > $c$.
From what I understand, the way to go is contour integration and you can make an argument that will allow you to use Jordan's lemma to simplify the expression (honestly, I'm unable to justify it). The square root introduces a branch cut on the imaginary axis from $x = ib$. The square root in the exponential is really confusing me.
I found a close question here asked a year ago, but so far no answers: How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?
Any help would be appreciated!
UPDATE: I also see a related technique here on pages 14 and 15: http://higgs.physics.ucdavis.edu/QFT-I.pdf
Here the author is able to estimate the integral and show it is non-zero. Still doesn't help me because I don't understand stationary phase techniques, but maybe someone sees something I don't.