I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.
We have given $f:\mathbb{R}^d \rightarrow \mathbb{R}$ with $f(x) = e^{-\frac{1}{2}\mid x\mid^2}$.
Show that the fourier trafsform $\hat f $ is given with $(\sqrt{2 \pi})^d f$.
So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.
We defined the fourier transform of $f \in L^1(\mathbb{R^d,\mathbb{C}})$ as
$\hat f(\xi) := \displaystyle\int_{\mathbb{R}^d}f(x)e^{-i\xi \cdot x} \mu(dx)$
also we know we following:
$\partial_j\hat f(\xi) = -i\displaystyle\int_\mathbb{R^d} x_j f(x)e^{-i\xi \cdot x} \mu(dx)$ if $(x_1,...,x_d) \mapsto x_j f(x)$ is integrable for every $j \in \lbrace 1,...,d \rbrace$.
Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.
So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $\hat f$ which provides the solution $\hat f = \sqrt{2 \pi} f$.
I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $\hat f$ but using a method from complex analysis which Im not allowed to use.