# Fourier Transform

## Homework Statement

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

I need to find the Fourier Transform of these integrals.

## Homework Equations

When looking at this expression, the integrals on the right are evaluated first then proceed to the left.

## The Attempt at a Solution

$$After applying the Fourier transform to the integral, we obtain: \begin{subequations} \begin{eqnarray} i\omega \tilde{\hat{{\cal E}}}(\omega) &=& -\kappa \tilde{\hat{{\cal E}}}(\omega) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})} \nonumber\\ & & -\frac{g^{2}}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')} \nonumber\\ & & + \sqrt{2\kappa}\, \tilde{\hat{{\cal E}}}_{in}(\omega), \\ \nonumber &=& -\kappa \tilde{\hat{{\cal E}}}(\omega) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})} \nonumber\\ & & -\frac{g^{2}}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')} \nonumber\\ & & + \sqrt{2\kappa}\, \tilde{\hat{{\cal E}}}_{in}(\omega), \\ \nonumber \end{eqnarray} \end{subequations}$$