User:Caliburn/s/6/Proof 1
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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.
Proof
Note that for each $z \in \C$, we have:
- $\size {\map \Re {\map f z} } = \size C < 2 \size C$
So:
From Entire Function with Bounded Real Part is Constant, we have:
- $f$ is constant.
$\blacksquare$