Complex Cosine Function is Entire/Proof 2

Proof
Let:


 * $\map f z = \exp z$


 * $\map g z = i z$


 * $\map h z = -i z$

for all $z \in \C$.

By Complex Exponential Function is Entire, we have that $f$ is entire.

By Polynomial is Entire, we have that $g$ and $h$ are entire.

Therefore, by Composition of Entire Functions is Entire, we have that both $f \circ g$ and $f \circ h$ are entire.

By Linear Combination of Entire Functions is Entire, we then have that:


 * $\displaystyle \frac 1 2 \paren {f \circ g + f \circ h}$

is entire.

Note that:

Therefore, $\cos$ is an entire function.