2. Complex Numbers

Magnitude: $|z|=\sqrt{a^2+b^2}$

Triangle inequality: $|z+w|\le |z|+|w|$

Arithmetic Rules

• $z\cdot \bar{z}=|z{|}^2$
• $z+\bar{z}=2\mathrm{Re}(z)$ and $z-\bar{z}=2i\mathrm{Im}(z)$
• $\overline{\stackrel{ˉ}{z}}=z$
• $\overline{z\pm w}=\bar{z}\pm \bar{w},$ for $z,w\in \mathbb{C}$
• $\overline{z\cdot w}=\bar{z}\cdot \bar{w},$ for $z,w\in \mathbb{C}$
• $z=\bar{z}$, if $z\in \mathbb{R}$
• $\bar{z}=-z$, if $z$ only imaginary
• $\frac{z}{w}=\frac{z\cdot \bar{w}}{|w{|}^2}$

Polarform

Every complex num is uniquely characterized by it’s distance from the origin in the complex plane and the angle $\phi$:

$$\begin{array}{rl}z& =a+ib\\ & =r\cdot (\cos a+i\sin b)\\ & =r\cdot {\mathrm{e}}^{\mathrm{i}\phi }\end{array}$$

Theorem

For $n\in \mathbb{N}$, the solutions of the equation $z^n=r{e}^{i\phi }$ are given by

$$z_k=r^{1/n}{e}^{i(\phi /n+2k\pi /n)},k=0,\dots ,n-1$$

Euler’s Formula

$e^{it}=\cos (t)+i\sin (t)$