Mathematik-Online problems: Problem 8: Subsets of the Complex Numbers, Complex-Valued Function

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	Mathematik-Online problems:
	Problem 8: Subsets of the Complex Numbers, Complex-Valued Function

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Let be $w, u \in \mathbb{C}$ , $a, b, c \in \mathbb{R}$ and $a \neq 0$ .

a)

Describe the following subsets of $\mathbb{C}$ :

i) $M_1 = \{z\in \mathbb{C} \mid az\bar{z} + z\bar{w} + \bar{z}w+b = 0\}$

ii) $M_2 = \{z\in \mathbb{C} \mid z\bar{u} + \bar{z}u+c = 0\}$

b)

Show that the map

$\displaystyle f: \, \mathbb{C}\setminus\{0\} \longrightarrow \mathbb{C}\setminus\{0\}, \qquad z \longmapsto \frac{1}{z}$

maps lines and circles onto lines and circles, if necessary without origin.

(Authors: Kimmerle/Roggenkamp/Höfert)

Solution:

automatisch erstellt am 15. 10. 2004

i)	$M_1 = \{z\in \mathbb{C} \mid az\bar{z} + z\bar{w} + \bar{z}w+b = 0\}$
ii)	$M_2 = \{z\in \mathbb{C} \mid z\bar{u} + \bar{z}u+c = 0\}$