Chapter 1: Complex numbers Flashcards
Common triangle inequality
$|a+b| \leq |a|+|b|$
Less common triangle inequality
$|a| - |b| \leq ||a|-|b|| \leq |a-b| $
$z \bar z$
$|z|^2$
$|\bar z|$
$|z|$
$z=x+iy$ in polar form
$x = ? \ y=?$
$x=r \cos{\theta}$
$y=r \sin{\theta}$
z in polar form
$z = r e^{i \theta}$
Arg z
The principal value of arg z
The unique value of $\theta$ that lies in the interval $(-\pi , \pi]$
arg($z_1 z_2$) = ?
arg $z_1$ + arg $z_2$
arg$\left ( \frac{z_1}{z_2} \right ) =$?
arg $z_1$ - arg $z_2$
$( \cos{\theta} + i \sin{\theta})^n = $
$\cos{n\theta} + i \sin{n \theta}$
because $(e^{i \theta})^n = e^{i n \theta}$
Roots of a complex number
Given $0 \not = z_0 \in \mathbb{C} $
where $$ z_0 = r_0 e^{i \theta_0}$$
then $z_0$ has n distinct nth roots given by
$ c_k = \sqrt[n]{r_0}$ exp $\left [ i \left ( \frac{\theta_0 +2k \pi}{n} \right ) \right ]$ for k = 0, 1, …, n-1
Where $\sqrt[n]{r_0}$ is the unique positive nth root of the positive real number $r_0$