Why is $ \overline{e^z} = e^\overline{z} $?

How can you conjugate an entire function? $ \overline{exp(z)} $ I need an equivalent.

I thought this is only possible with complex numbers.

What is the proof for $ \overline{e^z} = e^\overline{z} $ ? (Please don't involve a power series here.)

$$\overline{e^z}=\overline{e^xe^{iy}}=e^x(\overline{\cos y+i\sin y})=e^x(\cos y-i\sin y)=e^x(\cos(-y)+i\sin(-y))=e^{x-iy}=e^{\overline{z}}$$

Using another definition. $$ e^z = \sum_{k=0}^\infty \frac{1}{k!}\;z^k \\ \overline{e^z} = \overline{\sum_{k=0}^\infty \frac{1}{k!}\;z^k} = \sum_{k=0}^\infty \overline{\;\frac{1}{k!}\;}\;\overline{z^k} = \sum_{k=0}^\infty \frac{1}{k!}\;{\overline{z}\;}^k = e^{\overline{z}} $$