User:Caliburn/s/6

Theorem
Let $f : \C \to \C$ be an entire function.

Let the real part of $f$ be constant.

That is, there exists a positive real number $C$ such that:


 * $\map \Re {\map f z} = C$

for all $z \in \C$, where $\map \Re {\map f z}$ denotes the real part of $\map f z$.

Then $f$ is constant.