Fourier Transform Homework: Solving P(t) with E(t_1) & E(t_2)

[Niles]

Homework Statement

Hi

I wish to Fourier transform the following expression

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

What I do is the following

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

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 each other. But do you have a hint for what I need to do from here?

Cheers,
Niles.

 

[Niles]

Ok, so what we have is

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

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