Complex Sets 1. Sketch the following sets and determine which are domains: (a) |z2+i| ≤ 1; (b) |2z+ 3 > 4; (c) Im z > 1; (d) Im z = 1; (e) 0 ≤ arg z ≤ π/4 (z = 0); (f) |z4| ≥ |z|.
Complex Sets 1. Sketch the following sets and determine which are domains: (a) |z2+i| ≤ 1; (b) |2z+ 3 > 4; (c) Im z > 1; (d) Im z = 1; (e) 0 ≤ arg z ≤ π/4 (z = 0); (f) |z4| ≥ |z|.
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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![Complex Sets
1. Sketch the following sets and determine which are domains:
(a) |z – 2 + i| < 1;
(c) Im z > 1;
(e) 0 < arg z < /4 (z # 0);
(b) |2z + 3| > 4;
(d) Im z = 1;
(f) \z – 4| > |z].
2. Which sets in Exercise 1 are neither open nor closed?
3. Which sets in Exercise 1 are bounded?
4. In each case, sketch the closure of the set:
-T < arg z <I (z # 0);
(b) |Re z| < ]z\;
1
1
(c) Re
(d) Re(z?) > 0.
5. Let S be the open set consisting of all points z such that |z| < 1 or |z – 2| < 1. State
why S is not connected.
6. Show that a set S is open if and only if each point in S is an interior point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e56f833-52fb-457a-9d9f-48a97bbbd4fe%2Fa2e00ee5-200f-4ff6-adc6-8a48b8dc1c24%2Fau0uqn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Complex Sets
1. Sketch the following sets and determine which are domains:
(a) |z – 2 + i| < 1;
(c) Im z > 1;
(e) 0 < arg z < /4 (z # 0);
(b) |2z + 3| > 4;
(d) Im z = 1;
(f) \z – 4| > |z].
2. Which sets in Exercise 1 are neither open nor closed?
3. Which sets in Exercise 1 are bounded?
4. In each case, sketch the closure of the set:
-T < arg z <I (z # 0);
(b) |Re z| < ]z\;
1
1
(c) Re
(d) Re(z?) > 0.
5. Let S be the open set consisting of all points z such that |z| < 1 or |z – 2| < 1. State
why S is not connected.
6. Show that a set S is open if and only if each point in S is an interior point.
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