Chapter 1: Complex numbers

Question 1

Common triangle inequality

Answer 1

$|a+b| \leq |a|+|b|$

Question 2

Less common triangle inequality

Answer 2

$|a| - |b| \leq ||a|-|b|| \leq |a-b| $

Question 3

$z \bar z$

Answer 3

$|z|^2$

Question 4

$|\bar z|$

Answer 4

$|z|$

Question 5

$z=x+iy$ in polar form

$x = ? \ y=?$

Answer 5

$x=r \cos{\theta}$

$y=r \sin{\theta}$

Question 6

z in polar form

Answer 6

$z = r e^{i \theta}$

Question 7

Arg z

Answer 7

The principal value of arg z

The unique value of $\theta$ that lies in the interval $(-\pi , \pi]$

Question 8

arg($z_1 z_2$) = ?

Answer 8

arg $z_1$ + arg $z_2$

Question 9

arg$\left ( \frac{z_1}{z_2} \right ) =$?

Answer 9

arg $z_1$ - arg $z_2$

Question 10

$( \cos{\theta} + i \sin{\theta})^n = $

Answer 10

$\cos{n\theta} + i \sin{n \theta}$

because $(e^{i \theta})^n = e^{i n \theta}$

Question 11

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

Answer 11

$ 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 $r_0$