Cycles of real functions and the derivative operator If you start with $\sin(x)$ and take four derivatives, you get three distinct functions and then end up back at $\sin(x)$. For $e^{-x}$, you take two. For $e^{x}$, you only need to take one.
Generally, I will call an N-cycle of the derivative operator a sequence of real valued functions of a real variable including $f(x)$ where,
$$
\frac{d^Nf}{dx^N}=f(x)
$$
but,
$$
\frac{d^mf}{dx^m}\neq f(x)
$$
for every $m$ that is not a multiple of $n$.
Over the complex numbers, these functions are simply $\exp( x\sqrt[N]1)$. For $N=1,2,4$ there are cycles of real functions, which in the case of $N=4$ may be constructed as a linear combination of the complex functions. Can there be cycles of real functions of length other than 1,2 or 4? If so, how may they be obtained?
 A: Take the real or imaginary part of a complex cycle.  Thus if $\alpha + \beta i$ is a complex $n$'th root of unity, take
$$f(x) = \text{Re}(\exp((\alpha+i \beta) x)) = \exp(\alpha x) \cos(\beta x)$$
and you'll have $$f^{(k)}(x) = \text{Re}((\alpha +\beta I)^k \exp((\alpha + I\beta) x))$$
and in particular $f^{(n)} = f$.
A: Cycles of every length exist. Consider the solutions to the differential equation $f^{(n)}(x) -f(x)=0$. The solutions of this form a $n$-dimensional vector space, all of which are either $n$-cycles or cycles of some shorter length $d$ dividing $n$. Also, every $n$-cycle is a solution to this differential equation, by definition.
So then, for each $d|n$, the cycles of length dividing $d$ form a subspace of dimension at most $d$ in this larger vector space. A union of finitely many proper vector subspaces over $\mathbb{R}$ cannot be the whole space, and there is something in the space which has minimum cycle length $n$. Done.
We can also find solutions explicitly. If $\omega = \alpha +\beta i$ is a primitive $n$-th root of unity, $e^{\alpha x}\cos(\beta x)$ and $e^{\alpha x}\sin(\beta x)$ are $n$-cycles. This comes from using the similarity between constant-coefficent differential equations and polynomial equations - we factor the "polynomial" differential operator $D^n-1$, and the factor corresponding to the $n$th cyclotomic polynomial gives us $n$-cycles.
