Mathematik-Online problems: Problem 7: Sketching of Subsets of the Complex Numbers

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	Mathematik-Online problems:
	Problem 7: Sketching of Subsets of the Complex Numbers

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

Find the following subsets of $\mathbb{C}$ and make a sketch of them in the Gauß plane:

a) $\{z\in\mathbb{C}\mid \vert z-{\mathrm{i}}\vert < 2\}$      b) $\{z\in\mathbb{C}\mid {\mathrm{Re}}\,(z) \cdot {\mathrm{Im}}\,(z) \geq 1\}$

c) $\{z\in\mathbb{C}\mid z+\bar{z} -2 \leq -{\mathrm{i}}(z-\bar{z}) \leq z+\bar{z}+2\}$      d) $\{z\in\mathbb{C}\setminus\{0\}\mid {\mathrm{Re}}\,(\frac{1}{z}) \geq 1\}$

e) $\{z\in\mathbb{C}\mid \vert z\vert \leq \arg\,(z)\}$      f) $\{z\in\mathbb{C}\mid \vert z\vert < \vert z+3\vert\}$

g) $\{z\in\mathbb{C}\mid 1 < \vert z-{\mathrm{i}}\vert < 2 \, \wedge \, \vert\arg(z) -\frac{\pi}{2}\vert\leq \frac{\pi}{4} \}$

(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

Keyword: Numbers: complex

[Solutions]

automatisch erstellt am 29. 10. 2004

a) $\{z\in\mathbb{C}\mid \vert z-{\mathrm{i}}\vert < 2\}$		b) $\{z\in\mathbb{C}\mid {\mathrm{Re}}\,(z) \cdot {\mathrm{Im}}\,(z) \geq 1\}$
c) $\{z\in\mathbb{C}\mid z+\bar{z} -2 \leq -{\mathrm{i}}(z-\bar{z}) \leq z+\bar{z}+2\}$		d) $\{z\in\mathbb{C}\setminus\{0\}\mid {\mathrm{Re}}\,(\frac{1}{z}) \geq 1\}$
e) $\{z\in\mathbb{C}\mid \vert z\vert \leq \arg\,(z)\}$		f) $\{z\in\mathbb{C}\mid \vert z\vert < \vert z+3\vert\}$
g) $\{z\in\mathbb{C}\mid 1 < \vert z-{\mathrm{i}}\vert < 2 \, \wedge \, \vert\arg(z) -\frac{\pi}{2}\vert\leq \frac{\pi}{4} \}$