Complex Exponentials
Table of Contents
1. Complex Exponentials
Consider the following complex exponential with the following properties:
\begin{align} x: \mathbb{R} &\rightarrow \mathbb{C} \notag \\ x(t) &= e^{it} \notag \\ x(0) &= 1 \notag \\ \dot{x}(t) &= ie^{it} \notag \end{align}We claim this is actually just the unit circle in the complex plane. Consider the function as a "particle," with its time derivative (velocity) plotted below:
But, this is just circular motion! Additionally, since the speed of the particle is just the magnitude of the velocity, we have:
\begin{align} |\dot{x}(t)| = |ie^{it}| = 1 \notag \end{align}1.1. Euler's Formula
With this knowledge, we can represent the position of the particle at time \(t\) like so:
Since the speed of the particle is \(1\), at time \(t\) the particle would have moved \(t\) radians away from the center. With trigonometry, it is easy to see that we can represent this point as \(\cos t + i\sin t\). This yields the famous Euler's formula:
\begin{align} \boxed{e^{it} = \cos t + i \sin t} \end{align}Plugging \(-t\) into the formula,
\begin{align} e^{-it} = \cos t - i\sin t \notag \end{align}Adding and subtracting this with the original formula yields us the inverse Euler's formulas:
\begin{align} \boxed{\cos t = \frac{e^{it} + e^{-it}}{2}} \\ \boxed{\sin t = \frac{e^{it} - e^{-it}}{2i}} \end{align}