Complex Analysis/Elementary Functions/Exponential Functions

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Consider the real-valued exponential function ex. 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 ez has been already defined for z = iθ and we have eiθ = cosθ + isinθ.

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