Complex Analysis/Elementary Functions/Exponential Functions

< Complex Analysis‎ | Elementary 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.