Integral of form e^(x)/(x^2+a^2)

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Homework Statement


I'm working on converting a single-dimension wavefunction to its momentum representation. Here is the integral I am stuck with (I've pulled out some constants):
\int\limits_{-\infty}^{\infty}\frac{e^{\frac{-ipx}{\hbar}}}{x^2+a^2}\textrm{d}x


Homework Equations


Integration by parts, x = atanθ, e^iθ = cosθ + isinθ


The Attempt at a Solution


I've tried integrating by parts, but the problem is I always get something nasty multiplied by an exponential and I can't seem to make them get along. I've also tried Euler's formula, but that doesn't seem to help me either (I end up with something like cos(tanθ)dθ after trying to make a trig substitution to get the 1/(x^2 + a^2) part to behave.
Any ideas?
 
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Use the methods of complex integration and the residue theorem by taking a semi-circular contour depending on the sign of p.
 
Apparently, where I come from, rumor has it that you can't study Fourier analysis* without going through complex analysis before.

*and its applications to physical sciences.
 
I'm a grad student in ECE and it's been about 5 years since I took a complex analysis course. Unfortunately, I didn't use any of it after the class, so I'm more than a little rusty. Time to whip out some old textbooks and start digging.
Thanks!
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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