Let be a complex number, so .
It means we can write as
Now because as well, we can write as
in its turn.
If then and we can see
With en we now know that in on the unity circle. It means we can write as polar coordinates:
Now what can be said about and ? Let’s assume we can differentiate in as we can in :
So both derivatives behave the same
And as is for both functions, it means that the graphs and thus functions are the same:
For we get
called Euler’s identity, according to many the most beautiful formula, as it contains , , , as well as .