Complex Analysis/Elementary Functions/Exponential Functions

Consider the real-valued exponential function $$exp : \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$ exp(x) = e^x $$. It has the following properties:

1) $$e^x\neq 0\quad \forall x\in \mathbb{R}$$

2) $$e^{x+y} = e^xe^y\quad \forall x,y\in\mathbb{R}$$

3) $$(e^x)' = e^x\quad \forall x\in\mathbb{R}$$

We want to extend the exponential function $$exp$$ to the complex numbers in such a way that

1) $$e^z\neq 0\quad \forall z\in \mathbb{C}$$

2) $$e^{z+w} = e^ze^w\quad \forall z,w\in\mathbb{C}$$

3) $$(e^z)' = e^z\quad \forall z\in\mathbb{C}$$

But $$e^z$$ has been already defined for $$z=i\theta$$ and we have $$e^{i\theta}=\cos \theta+i\sin\theta$$.