What is the fourier transform of this function?

Question

With $$ f(x) = \frac{1}{p} e^{-\pi x^2/p^2} $$ and $p>0$, I got an answer of $\displaystyle e^{-\pi p^2u^2}$.

I just wanted to make sure I got the right answer. If I didn't, I will work through the problem and try again. Thanks.

Answer 1

Yes, the answer is correctly derived from the scaling property of the Fourier transform. The function $\exp(-\pi^2 x^2/p^2)$ is $\exp(-\pi x^2)$ scaled by $1/p$. Therefore, the transform scales by $p$ and acquires a factor of $p$, becoming $p\exp(-\pi^2 p^2 x^2)$. Then divide by $p$ throughout.