Fourier Transform

1. Dec 7, 2008

Nusc

1. The problem statement, all variables and given/known data
I need to find the Fourier Transform of this integro-differential equation:

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

2. Relevant equations
$$\hat{{\cal E}}}(t)$$
is just a function of t
3. The attempt at a solution
$$After applying the Fourier Transform, \begin{subequations} \begin{eqnarray} \omega \; \tilde{\hat{{\cal E}}}(\omega) &=& -\frac{\kappa}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{{\cal E}}(t)e^{-i\omega t} dt + ig\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\, \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})} \nonumber\\ & & - g^{2}e^{-(\gamma +i\Delta)(t-t_{0})}\int_{-\infty}^{\infty} d \Delta\; \hat{{\cal \rho}}(\Delta)\,\int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{(\gamma +i\Delta)(t-t')} + \sqrt{2\kappa}\, \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{{\cal E}}_{in}(t)e^{-i\omega t} dt, \nonumber\\ \nonumber \end{eqnarray} \end{subequations}$$

is this correct?

Last edited: Dec 8, 2008
2. Dec 8, 2008

Nusc

Is this equation too intimidating?

3. Dec 8, 2008

Nusc

Any thoughts? Did I not make myself clear?

4. Dec 8, 2008

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.