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EM: Fourier Transform

  1. Oct 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Hi

    I wish to Fourier transform the following expression

    [tex]
    P(t) = \int\limits_{ - \infty }^\infty {dt_1 dt_2 \chi (t - t_1 ,t - t_2 )E(t_1 )E(t_2 )}
    [/tex]

    What I do is the following

    [tex]
    \int\limits_{ - \infty }^\infty {P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 (t - t_1 )} e^{ - i\omega _2 (t - t_2 )} e^{ - i\omega _1 t_1 } e^{ - i\omega _2 t_2 } }
    [/tex]

    I'm pretty sure I need to keep rewriting the expressions on the LHS and RHS until I reach a point, where I can compare the terms to eachother. But do you have a hint for what I need to do from here?

    Cheers,
    Niles.
     
  2. jcsd
  3. Oct 5, 2011 #2
    Ok, so what we have is

    [tex]
    \int\limits_{ - \infty }^\infty {d\omega P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 (t - t_1 )} e^{ - i\omega _2 (t - t_2 )} e^{ - i\omega _1 t_1 } e^{ - i\omega _2 t_2 } }
    [/tex]
    [tex]
    \int\limits_{ - \infty }^\infty {d\omega P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 t} e^{ - i\omega _2 t}}
    [/tex]

    But this seems a little odd, because what am I supposed to do about the integral over t1 and t2?
     
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