Math 303 - Complex Variables
Homework due January 28
Question 1. [This is derived from questions 2 - 7 in Section 1.6 of Saff-Snider]. Consider the sets
described by the following inequalities.
(i) |z−1 + i| ≤ 3
(ii) |Arg z|< π/4
(iii) 0 <|z−2|<3
(iv) −1<Im z≤1
(v) |z| ≥ 2
(vi) |Re z|2>1
Answer the following questions about the above sets:
(a) Sketch each of the given sets.
(b) Which of the given sets are open? Explain why.
(c) Which of the given sets are domains? Explain why.
(d) Describe the boundary of each of the given sets.
Question 2. Let Sbe the set of points 1,1
2,1
3,1
4, . . ..
(a) Draw the set Son the complex plane.
(b) What is the boundary of S?
(c) Is this set connected? Explain why or why not.
(d) Is this set open? Is this set closed? Explain why or why not.
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