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

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

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 $e^{i\theta }=\cos \theta +i\sin \theta$.