I am currently looking at an example of how to calculate the Fourier Transform for the function \begin{equation} f(x) = \exp\left({-\frac{x^2}{2}}\right) \end{equation} Now $f$ solves the differential equation \begin{equation} f'(x) = -xf(x) \end{equation} and so, applying the FT to both sides gives \begin{equation} i \xi\,\hat{f}(\xi) = -i(\hat{f})'(\xi) \end{equation} Using these two equations we can derive that \begin{equation} \left(\frac{\hat{f}(x)}{f(x)}\right)' = 0 \end{equation} hence \begin{equation} \hat{f}(x) = c_0f(x) = c_0\exp\left({-\frac{x^2}{2}}\right) \end{equation}
And here is where I have trouble, because the author of the notes that I am using says that $c_0 \geq 0$, which I can't see why .. I am sure it must be something obvious and I am just too blind to see it. Any hint would be highly appreciated, many thanks !
Edit: It is not the claim that I don't understand (I know it is right), it is just that I can't see why this is true looking solely at the derivations I have so far.