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Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number.

Are there such functions $f(z)$ where $\Re(f(z))$ is periodic in the domain (no periods larger than the domain please :p ) but $f(z)$ is not periodic? (if $D\subset \mathbb C$ it is clear that $f(z)$ is not periodic but $\Re(f(z))$ might still be for some shapes of $D$).

In particular the case when $D = \mathbb C$ is interesting. (in other words $f(z)$ meromorphic over $\mathbb C$)

I guess it is a similar question to ask about $\Im$ , $\operatorname{Arg}$ or $|\cdot|$ instead of $\Re$.

I read about double periodic functions and Cauchy-Riemann equations but I still don't know. I can't find such a function in the literature ( i mean the one i search here , i don't mean i can't find a double periodic one in the literature of course ) and I don't know how to construct them or even if they exist.

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The literature is literally filled with examples of doubly periodic functions. They are called elliptic functions. You can find them constructed in pretty much every textbook on complex analysis —Lang's, Ahlfors's, &c. – Mariano Suárez-Alvarez Sep 12 '12 at 19:59
I was referring to the functions i was looking for. Maybe an edit. – mick Sep 12 '12 at 20:01
Also, your parenthetical remark in the 2nd paragraph that something is clear is quite weird: there certainly are periodic functions defined on proper subsets of the complex plane; for example, restrict any periodic function from the plane to any open subset of the plane invariant under the periods. – Mariano Suárez-Alvarez Sep 12 '12 at 20:01
A function cannot repeat its period 2 times and then diverge when it is suppose to reach its 3rd repeat. At least not a function that is holomorphic in its period. – mick Sep 12 '12 at 20:09
I do not know what your last comment means. – Mariano Suárez-Alvarez Sep 12 '12 at 20:11
up vote 11 down vote accepted

Suppose $f$ is meromorphic in $\mathbb C$ and its real part is periodic of period $p$. Then $z\mapsto f(z)-f(z+p)$ is a meromorphic function whose real part is identically zero.

Can you conclude something from this?

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Of course ! So simple. Thank you. – mick Sep 12 '12 at 20:04

The real part of $z\mapsto iz$ has period $p$ for any $p\in \mathbb R$.

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