Fourier Transform

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Homework Statement


[tex]
\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}
[/tex]

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


[tex]
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}
[/tex]
 

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