Let be a complex number, so .

It means we can write as

So

Now because as well, we can write as

in its turn.

If then and we can see

or

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

And if

Then

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 .