Complex Analysis/Elementary Functions/Exponential Functions

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

1) $e^x\ne 0\mathrm{\forall }x\in ℝ$

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

3) $(e^x{)}^{\prime }=e^x\mathrm{\forall }x\in ℝ$

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

1) $e^z\ne 0\mathrm{\forall }z\in ℂ$

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

3) $(e^z{)}^{\prime }=e^z\mathrm{\forall }z\in ℂ$

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