I analyse the proof that the derivative of the function below exists and is zero, and that the function is constant itself. Let's define $$f(u) = \int_{-\infty}^\infty e^{{u^2\over2}-{x^2\over2}+iux} dx$$ Now the proof goes as follows: $$f'(u) = \int_{-\infty}^\infty {d \over du} e^{{u^2\over2}-{{x^2 \over 2}+iux}} dx = \int_{-\infty}^\infty (u+ix)e^{{u^2\over2}-{{x^2 \over 2}+iux}} dx = f(u)|_{-\infty}^{\infty} = 0-0 = 0$$
Since the derivative is zero, the function itself is constant.
Now the proof is quite simple, but there is one problem here - to use the Liebniz integral rule it needs to be that $(u+ix)e^{{u^2\over2}-{{x^2 \over 2}+iux}}$ can be absolutely bounded by a function of $x$ only (so the Leibniz integral rule can be used). My professor said this is possible to rewrite the function in such a way that a bound is easily found, but I have been struggling for many hours unable to find one, and at this point I am not even sure if it exists (and it's one of those "it's easy to see..." remarks). Any help would be greatly appreciated.