What Strategies Can Tackle This Complex Contour Integral?

[MadMax]

Contour integral

How would you deal with this?

$$\int \frac{\rho \sin{\theta} d \rho d \theta}{\cos{\theta}} \frac{K^2}{K^2 + \rho^2} e^{i \rho \cos{\theta} f(\mathbf{x})}$$

if the cos(theta) were'nt on the bottom I'd have no problem; I'd simply substitute for cos(theta) and the sin(theta) would cancel...

but as it stands.. I'm stumped.

Help/hints would be much appreciated. Thanks for reading.

 

[Dick]

Are you sure there should be a cos(theta) in the denominator? Where did that expression come from?

 

[MadMax]

well I had something like:

$$\int \frac{d^3 \mathbf{q}}{\mathbf{q_z}} \frac{2K^2 + \mathbf{q}^2 - (\mathbf{q} \cdot \hat{x})^2}{2(K^2 + \mathbf{q}^2)}e^{i \mathbf{q_{\perp}} \cdot \mathbf{x}}e^{i \mathbf{q_z} f(\mathbf{x})}$$

I made q_z parallel to x, (which means the first exponential disappears), converted into spherical coords, integrated over \phi from 0 to 2pi, and thus...

 

[Dick]

I guess I'm stumped as well. Anybody else seen something like that?

 

[MadMax]

I wasn't correct in some of the things I wrote in the two equations. The second exponential in particular...

I edited them so they're correct now. Not sure if it makes a difference but perhaps it helps? Might make sense how I got from one to the other now anyway...