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

[zje]

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?

 

[Dickfore]

Use the methods of complex integration and the residue theorem by taking a semi-circular contour depending on the sign of p.

 

[dextercioby]

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.

 

[zje]

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!