Euler’s Identity

Let z be a complex number, so z\in\mathbb{C}.
It means we can write e^z as



    \[e^{a+ib}=e^a\cdot e^{ib}\]

Now because e^{ib}\in\mathbb{C} as well, we can write e^{ib} as


in its turn.
If e^{ib}=x+iy then e^{-ib}=x-iy and we can see




With e^{ib}=x+iy en x^2+y^2=1 we now know that e^{ib} in on the unity circle. It means we can write e^{ib} as polar coordinates:


Now what can be said about b and \theta? Let’s assume we can differentiate in \mathbb{C} as we can in \mathbb{R}:



    \[f'(b)=e^{ib}\cdot i=ie^{ib}\]

And if




So both derivatives behave the same


And as f(0) is 1 for both functions, it means that the graphs and thus functions are the same:


For \theta=\pi we get


called Euler’s identity, according to many the most beautiful formula, as it contains e, \pi, i, 1 as well as 0.