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at the moment I am working on the following exercises:

Let $v,w \in \mathbb{R}^2 = \mathbb{C}$ be two linear independent $\mathbb{R}$-vector. Now let be $f:\mathbb{C} \to \mathbb{C}$ holomorphic and for $f$ holds the following equation: $$f(z)=f(z+v)=f(z+w).$$

Show that $f$ is constant.

Because of Liouville`s theorem (complex analysis) I know that a holomorphic function which is bounded is constant. For using this theorem, I need to show that $f$ is bounded.

But I have my problems to do that. Can someone give me a hint? I would appreciate that.

Alain Remillard
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WomBud
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  • Given any $z\in\mathbb{C}$ you can express $z$ as a linear combination of $v$ and $w$. Try to apply the periodicity of $f$ to find an element in a compact set with the same image as $z$. – Julio Puerta May 16 '24 at 12:58
  • $f$ is continuous and its behaviour is determined by its restriction on a compact set, so... – 2by2is2mod2 May 16 '24 at 12:58
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    Write $z=av + bw$ with $a,b\in \mathbb{R}$. Then we have $$ f(a v + bw)=f((a-\lfloor a \rfloor) v + (b-\lfloor b \rfloor) w). $$ Use this to show that $$f(\mathbb{C})= f({ cv +dw \ : \ c,d\in [0,1] }).$$ Use continuity and compactness to show $f(\mathbb{C})$ is bounded and conclude with Liouville. – Severin Schraven May 16 '24 at 13:06
  • Hello, thank you for your input I appreciate that. thanks to your suggestions i have a better understanding of the problem and could solve that. I am sorry for posting a multi-duplicate question. – WomBud May 16 '24 at 21:21

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Define the equivalence relation $z\sim \zeta$ if $z-\zeta=av+bw$ for some integers $a, b\in\mathbb{Z}.$ Note that when $z\sim\zeta$ we have $f(z)=f(\zeta).$ Therefore, to understand the range of this function, it is enough to consider it on the set of equivalence classes $\mathbb{C}/\sim.$

Next, if you look carefully this quotient set $\mathbb{C}/\sim$ can be identified with the quadrilateral $\mathcal{D}$ whose vertices are $0, v, w,$ and $v+w.$ Clearly, $\mathcal{D}$ is bounded + closed = compact.

Finally, consider the restriction $\vert f\vert :\mathcal{D}\to\mathbb{R}_{\ge0}.$ This is a continuous function on a compact set. Hence, there must be extreme values.

Bumblebee
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