9
$\begingroup$

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.

  • 0
    I thought reading about double periodic functions and Cauchy-Riemann equations would help , but its much simpler as you have shown. Oh well , it was fun reading anyway.2012-09-12

2 Answers 2

11

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?

  • 0
    Of co$u$rse ! So simple. Thank yo$u$.2012-09-12
1

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