just out of mathematical curiosity, I found out on some website that the following statement holds for any :
How would one go to actually prove this statement ? Without going in too advanced mathematics if possible.
You can integrate by parts n-times, or you can differentiate under the integral sign.
We know that
And notice that (1)
Now switch the order of integration and differentiation (which in this case is allowed).
Finally equate (1) and (2) and let a=1.