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