Fourier transform of exponential function

[Peter Alexander]

1. The problem statement, all variables, and given/known data
Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) = e^{-a t^2}$$

Homework Equations

The equation for Fourier transform, according to our textbook, is $$F(\omega) = \int_{-\infty}^{\infty} e^{i \omega t} f(t) d\omega$$ If I understand correctly, multiple notations exist, but this one was given to us.

The Attempt at a Solution

Normally the computation is pretty straightforward, simply plug $f(t)$ into the equation, perform some integration and done. However, this one gave me loads of troubles! A solution, according to output from Mathematica and our textbook solutions is $$F(\omega) = \sqrt{\frac{\pi}{a}} e^{-\frac{\omega ^2}{4 a}}$$ Some research later, I've found out that Fourier transform is linked to Dirac delta function, but this concept hasn't been discussed on lectures nor mentioned in the textbook. My suspicion is that link between exponential function and trigonometric functions (Euler's formula) can lead me a step closer to a solution.

Using Mathematica, I've seen that the integrand $e^{i \omega t} f_a(t)$ can be expressed as $$e^{t (-a t+i \omega )} = \cosh \left(a t^2-i t \omega \right)-\sinh \left(a t^2-i t \omega \right)$$ but since both integrals do not converge on $(-\infty, \infty)$ they cannot be computed. This claim was reafirmed when I tried integrading both functions in Mathematica.

At that point, I'm completely stuck. I thought about checking the given examples and trying to guess the value of Fourier transform this way, but even then I couldn't produce a solution. Would somebody be kind enough to help me?

 

[DrClaude]

You need to rescale time such that $f(t')$ gives you $f_a(t)$. Then rescale the frequency to $\omega'$ to recover the Fourier integral and then you can find $F(\omega')$ using the FT of the sample function.

Edit: To clarify what I meant, you want to find
$$
\int_{-\infty}^{\infty} e^{i \omega t} e^{-at^2} d\omega
$$
You know that
$$
\int_{-\infty}^{\infty} e^{i \omega t} e^{-t^2/2} d\omega = \sqrt{2\pi} e^{-\omega^2/2}
$$
so modify this equation by changing $t$ and $\omega$ until you get to the integral above.

 

[FactChecker]

Are you familiar with the scaling property of Fourier transformations? If so, you should use it (and maybe review the proof). You should become familiar with the use of any property that has been covered to this point in your class.

 

[Peter Alexander]

I actually solved it, can't believe it was that easy!
I would like to thank both of you for your time and effort helping me understand this problem.