# Expression for function's period

Is there expression for an operator that gives for any analytic periodic function its period?

P.S.

In my view this probably means solving the following system of equations:

$$f^{(n)}(0)=f^{(n)}(T)$$

against $T$.

I just wonder whether the solution to this system can be written in a form of one expression.

P.P.S. Alternatively the equation can be written as

$$\Delta f(Tz)≡0$$

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I can't think of any operator that would be able to handle both elliptic functions and the function $x-\lfloor x\rfloor$... – J. M. May 7 '11 at 8:28
What is an expression? What is a periodic function? (For example, is the indicator function of $\mathbb{Q}$ a periodic function? What is its period?) – Qiaochu Yuan May 7 '11 at 8:28
I am interested in operator that returns (minimal) period for analytic functions. – Anixx May 7 '11 at 8:57
So... it would have to return one result for exponentials and two results for elliptic functions, then? – J. M. May 7 '11 at 9:02
If you want to focus on analytic functions, do specify that. If you have further conditions on the functions you are interested in - please specify them. If you add context to why you want this operator, other ideas that might be useful can be given instead. – Asaf Karagila May 7 '11 at 10:08

If you just want an abstract operator you could define for the function $f: A \mapsto B$ the period length $P$ as (you need some norm to define whats "small"):

$P(f)=\min(p \in \mathbb{R} | \exists z: |z|=p: \forall x \in A: f(x+z)=f(x))$ if the minimum exists. If you want some generic function (that will be more useful) to actually determinate the period length you have to make some assumptions about $f$ as there probably isn't any more generic useful formula.

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Yes. I want a constructive formula that gives an actual expression or encodes an algorithm of how to find such period. – Anixx May 7 '11 at 13:04
As I said you have to make an assumption about the functions you want to look at then. – Listing May 7 '11 at 13:09
I meant analytic functions. – Anixx May 7 '11 at 13:10
So you can try to find the smallest strictly positive root (in terms of $y$) of $f(x)-f(x+y)$ which is again a numerical problem (note that it has to hold for all $x$). – Listing May 7 '11 at 13:13