Complex Analysis/Elementary Functions/Exponential Functions

Consider the real-valued exponential function $e^x$. 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 $e^{i\theta }=\cos \theta +i\sin \theta$.