---
title: "Euler's Equation"
source: https://www.jemoka.com/posts/kbheuler_s_equation/
---

\begin{equation} f(x) = e^{ix} = \cos (x) + i\sin (x) \end{equation}
this brings a circle of radius one, because in every point, velocity is orthogonal to where you are (because \(f&rsquo;(x) = if(x)\), and multiplying by \(i\) accounts for a rotation of 90 degrees.
And so,
\begin{equation} z = re^{i\theta} \end{equation}
gives any point in the imaginary polar plane.
