How do I tackle an integral like e^(i*w*t-t^2/T^2) from -infinity to infinity. This thing is supposed to be an fourier integral. I tried solving it with substitution and partial integration but nothing worked so far.

Question

anonymous · Answer

$$\int\limits_{-\infty}^{\infty} dt(e^{i \omega t-t^2/\tau^2})$$

anonymous · Answer

This is supposed to be G(omega) and I want to know what it is!!!

experimentx · Answer

this looks like Fourier transform of Gaussian fx e^(-t^2) ... just complete the square on the power and ... use that slight translation will not change the integral.

anonymous · Answer

So you want me to rewrite (a*t^2+bt) as (t-c)(t-d) and then do what exactly? I am sorry but I have no clue how to deal with this thing.

A crazy person who is sitting next to me wants to bring this stuff into an elliptical form because e^(-x^2) is integrated by a circle equation.

experimentx · Answer

No ... I want you to complete the square on the top as
at^2 + bt as at^2 + 2 b/sqrt(a) sqrt(a) t + 4 b^2 on the top.

experimentx · Answer

that crazy person might be right!! you can evaluate
$$ \int_{-\infty}^{\infty }e^{-x^2}dx$$
can be evaluated by changing into polar coordinates.

experimentx · Answer

there are other ways to evaluate it .. but that one is easiest.

anonymous · Answer

We got it (I guess)! Thank you, experimentX. I would never guessed to do it like that. I´ll try to remember that trick.