Fourier Transform of Integro-Differential Equation

[Nusc]

Homework Statement

I need to find the Fourier Transform of this integro-differential equation:

$$\begin{subequations}<br /> \begin{eqnarray}<br /> \nonumber<br /> \dot{\hat{{\cal E}}}(t) &=& -\kappa \hat{{\cal E}}(t) + i g\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\,( \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})} <br /> \nonumber\\<br /> & & + e^{-(\gamma + i\Delta)(t-t_{0})} ig\int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{(\gamma +i\Delta)(t-t')})<br /> \nonumber\\<br /> & & + \sqrt{2\kappa}\, \hat{{\cal E}}_{in}, \\<br /> \nonumber<br /> \end{eqnarray}<br /> \end{subequations}<br />$$

Homework Equations

$$\hat{{\cal E}}}(t)$$
is just a function of t

The Attempt at a Solution

$$After applying the Fourier Transform,<br /> \begin{subequations}<br /> \begin{eqnarray}<br /> \omega \; \tilde{\hat{{\cal E}}}(\omega) &=& -\frac{\kappa}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{{\cal E}}(t)e^{-i\omega t} dt + ig\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\, \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})}<br /> \nonumber\\<br /> & & - g^{2}e^{-(\gamma +i\Delta)(t-t_{0})}\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\,\int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{(\gamma +i\Delta)(t-t')}<br /> + \sqrt{2\kappa}\, \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{{\cal E}}_{in}(t)e^{-i\omega t} dt, \nonumber\\ \nonumber<br /> \end{eqnarray}<br /> \end{subequations}<br />$$

is this correct?

 

[Nusc]

Is this equation too intimidating?

 

[Nusc]

Any thoughts? Did I not make myself clear?

 

[Kaleb]

Honestly it looks ok, but something looks like it may be missing. I don't have much experience with it, but it does seem ok.