Functions with cyclic derivatives

Question

Possible Duplicate:
Functions that are their Own nth Derivatives for Real n

Popular function like sine, cos, Sinh, exp, etc. Have the property where for some fixed natural number k and all natural n

$y^{(nk)}=y$

Is the family limited in some sense? Is there a general solution to this whole set of differential equations?

This was more or less asked here, http://math.stackexchange.com/questions/7511/functions-that-are-their-own-nth-derivatives-for-real-n/7513#7513 — Mariano Suárez-Álvarez, Jun 22 '11 at 18:05

score 3 · Answer 1 · answered Jun 22 '11 at 18:01

3

These are all of the form $e^{ax}$ for $a$ being an $n^{\text{th}}$ root of $1$ or linear combinations of them. For example, $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$ and $i$ is a $4^{\text{th}}$ root of $1$

answered Jun 22 '11 at 18:01

Ross Millikan

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Functions with cyclic derivatives

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