Chapter 1: Complex numbers Flashcards

1
Q

Common triangle inequality

A

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

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2
Q

Less common triangle inequality

A

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

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3
Q

$z \bar z$

A

$|z|^2$

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4
Q

$|\bar z|$

A

$|z|$

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5
Q

$z=x+iy$ in polar form

$x = ? \ y=?$

A

$x=r \cos{\theta}$

$y=r \sin{\theta}$

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6
Q

z in polar form

A

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

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7
Q

Arg z

A

The principal value of arg z

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

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8
Q

arg($z_1 z_2$) = ?

A

arg $z_1$ + arg $z_2$

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9
Q

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

A

arg $z_1$ - arg $z_2$

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10
Q

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

A

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

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

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11
Q

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

A

$ 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 $r_0$

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