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Fourier Transform

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data
    I need to find the Fourier Transform of this integro-differential equation:

    [tex]
    \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}

    [/tex]


    2. Relevant equations
    [tex]
    \hat{{\cal E}}}(t)
    [/tex]
    is just a function of t
    3. The attempt at a solution
    [tex]
    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}

    [/tex]

    is this correct?
     
    Last edited: Dec 8, 2008
  2. jcsd
  3. Dec 8, 2008 #2
    Is this equation too intimidating?
     
  4. Dec 8, 2008 #3
    Any thoughts? Did I not make myself clear?
     
  5. Dec 8, 2008 #4
    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.
     
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