Exploring Interference of Plane Waves Using the Residue Theorem

[AwesomeTrains]

Homework Statement

I have to show that the interference of plane waves: [itex]f^{(\pm)}(\vec r,t)=\int \frac {d^3k}{(2\pi)^{3/2}}\int \frac {d\omega}{(2\pi)^{1/2}}e^{i(\vec k \cdot \vec r - \omega t)}\tilde f^{(\pm)}(\vec k, \omega)[/itex]

where the amplitudes are given as: [itex]\tilde f^{(\pm)}(\vec k, \omega)=\frac {2\delta(\omega-\omega_0)}{k^2-(\omega\pm i\delta)^2/c^2}[/itex]
is a spherical wave of the form: [itex]f^{(\pm)}(\vec r, t)=\frac{1}{r}e^{-i\omega_0(t\mp r/c)} [/itex]

Homework Equations

They recommend that I use the residue theorem.

The Attempt at a Solution

I thought about doing some sort of coordinate transformation.
What are the integration limits? They weren't given, do I have to figure those out?
Would it be useful to do a Fourier transform of the amplitudes?

Any tips to get me started are really appreciated. (I get confused when I look at the integral)

Alex

 

[AwesomeTrains]

Thought about it. I guess I should do the integral over [itex]d\omega[/itex] first, but what is the meaning of this [itex]\delta[/itex] in [itex](ω±iδ)^2/c^2[/itex] I know that it's the delta function when it has some argument but there it hasn't.