Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can anyone point me to a derivation of $x^n f(x)$? I know that the answer is $(i)^n$ times the $n$-th derivative of the transform of $f(x)$, but I've searched for a derivation and can't find it.

share|cite|improve this question
@ctype.h, every time you edit an old problem (and you've done dozens), you bump it onto the front page, where it takes up valuable real estate. Please limit yourself to, maybe, three of these a day. – Gerry Myerson Nov 22 '12 at 6:04
@GerryMyerson Sorry, I was not aware that it would be problematic. I will scale down the editing. – ctype.h Nov 22 '12 at 6:52
up vote 2 down vote accepted

It suffices to prove the case with $n=1$ and then apply induction.

Consider applying $d/ds$ to both sides of the definition (below) and then multiplying both sides by $i$.

$$ F(s)=\mathcal{F}\{f(x)\}(s)=\int_{-\infty}^\infty f(x)e^{-isx}dx$$

(We must have sufficient regularity on $f$ for this derivation to be valid.)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.