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2. Is the set D = | = {z = C | |z| = 5, |z − i| > 5} compact (C is equipped with a standard metric)?

2. Is the set D = | = {z = C | |z| = 5, |z − i| > 5} compact (C is equipped with a standard metric)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question 2:** Is the set \( D = \{ z \in \mathbb{C} \mid |z| = 5, |z - i| > 5 \} \) compact, where \(\mathbb{C}\) is equipped with a standard metric? **Explanation:** This question is asking about the compactness of a set within the complex plane. The set \( D \) consists of all complex numbers \( z \) such that the magnitude of \( z \) is 5, which represents a circle of radius 5 centered at the origin. Additionally, the condition \( |z - i| > 5 \) implies that these points \( z \) are outside of a circle of radius 5 centered at \( i \) (the imaginary unit on the complex plane). To determine if the set is compact, two properties must be checked under the standard metric: closedness and boundedness. In complex analysis, a set is compact if it is closed and bounded.
Transcribed Image Text:**Question 2:** Is the set \( D = \{ z \in \mathbb{C} \mid |z| = 5, |z - i| > 5 \} \) compact, where \(\mathbb{C}\) is equipped with a standard metric? **Explanation:** This question is asking about the compactness of a set within the complex plane. The set \( D \) consists of all complex numbers \( z \) such that the magnitude of \( z \) is 5, which represents a circle of radius 5 centered at the origin. Additionally, the condition \( |z - i| > 5 \) implies that these points \( z \) are outside of a circle of radius 5 centered at \( i \) (the imaginary unit on the complex plane). To determine if the set is compact, two properties must be checked under the standard metric: closedness and boundedness. In complex analysis, a set is compact if it is closed and bounded.
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