Complex Analysis/Elementary Functions/Exponential Functions

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

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

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

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

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

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

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

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

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

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