## Prove conjugation law for Exp(z) using only Exp(x + y) = Exp(x)Exp(y)

Starting with just the property $$E(x + y) = E(x)E(y)$$, one can prove quite a lot of the main properties of the exponential function on real numbers. For example, $$E(0) = 1$$, and $$E'(X) = E(X)$$, and $$E(nx) = E(x)^n$$ all follow from a straight-forward application of definitions.

To move this in to complex analysis and Euler’s formula, it seems to me that the key property that needs to be proved is $$E(\overline{z})$$ = $$\overline{E(z)}$$.

Is there a nice way to prove this that is in the spirit of this particular line of reasoning?

Edit: For example, Chapter 8 of Baby Rudin follows this reasoning, but resorts to the definition $$E(x) = \Sigma_{n\ge0}(\frac{z^n}{n!})$$ to prove the conjugation property.