Consider the real-valued exponential function exp:R→R{\displaystyle exp:\mathbb {R} \rightarrow \mathbb {R} } defined by exp(x)=ex{\displaystyle exp(x)=e^{x}} . It has the following properties:
1) ex≠0∀x∈R{\displaystyle e^{x}\neq 0\quad \forall x\in \mathbb {R} }
2) ex+y=exey∀x,y∈R{\displaystyle e^{x+y}=e^{x}e^{y}\quad \forall x,y\in \mathbb {R} }
3) (ex)′=ex∀x∈R{\displaystyle (e^{x})'=e^{x}\quad \forall x\in \mathbb {R} }
We want to extend the exponential function exp{\displaystyle exp} to the complex numbers in such a way that
1) ez≠0∀z∈C{\displaystyle e^{z}\neq 0\quad \forall z\in \mathbb {C} }
2) ez+w=ezew∀z,w∈C{\displaystyle e^{z+w}=e^{z}e^{w}\quad \forall z,w\in \mathbb {C} }
3) (ez)′=ez∀z∈C{\displaystyle (e^{z})'=e^{z}\quad \forall z\in \mathbb {C} }
But ez{\displaystyle e^{z}} has been already defined for z=iθ{\displaystyle z=i\theta } and we have eiθ=cosθ+isinθ{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }.