I am currently computing the effect of an arbitrary driving on the dynamics of photons in a waveguide. Along the way I find the following integral equation for the complex function $F(\omega,\omega_q)$: $$ \delta(\omega_k-\omega_q)=\int_{-\infty}^{\infty}d\omega\frac{F(\omega,\omega_q)}{\omega-\omega_k+i0^+}e^{i\omega t_0}, $$ for any real and positive $\omega_k$ and $\omega_q$, where $\delta(x)$ is the Dirac delta. The parameter $t_0$ is real and can take in principle whatever value makes the integral easier.
This equation seemed not too difficult at the beginning, but I after days of attempts I feel it is beyond my expertise. I have tried many ways of solving it, basically assuming different forms for the function $F(\omega,\omega_q)$ that seemed suitable, but this has led nowhere so far. Direct application of the Sokhotski-Plemelj formula leads to no simplification as far as I can see.
Seeing that this equation does not look that difficult for an expert, I come to you for help. Has anyone encountered such equation before, and does anyone know how to solve it (or if it is even possible)?
Thank you very much in advance!