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:

the real part of $f$ is bounded.

From Entire Function with Bounded Real Part is Constant, we have:

$f$ is constant.

$\blacksquare$