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MAT334 – Complex Variables

Fall 2023

Homework Assignment 5

Due Friday, November 24, 2023

Grading

• The total nominal mark of this homework is 40 points. If you receive x ∈ [0, 40] points, then your grade G will be recorded as

Thus, you do not need to answer correctly all the questions to obtain a perfect grade. However, you are strongly encouraged to work on as many questions as you can – if not all of them.

• Late submissions will normally not be accepted and will be marked 0.

Notation. Given w ∈ C and r ∈ R>0, we define

D(w, r) = {z ∈ C : |z − w| < r} and D(w, r) = {z ∈ C : |z − w| ≤ r}

to be the open and closed discs with centre w and radius r.

Exercise 1 (8 points)

Consider the function f : C \ {0, 1} → C defined by

Question 1 (2 points). Consider the curve α: [0, 2π] → C defined by α(t) = −2023 + 11e it . Compute Rαf(z) dz.

Question 2 (2 points). Consider the curve β : [0, 2π] → C defined by β(t) = −e it/2. Compute Rβf(z) dz.

Question 3 (2 points). Consider the curve γ : [0, 2π] → C defined by γ(t) = 1 + e iπ/3 + √ 2e it . Compute Rγf(z) dz.

Question 4 (2 points). Prove that f has no antiderivative on C \ {0, 1} – that is, show that there is no holomorphic function F : C \ {0, 1} → C such that F′ = f.

Exercise 2 (10 points)

Consider the smooth closed curve γ and the domains D1, D2, D3, D4 ⊂ C represented on the figure below.

Question 1 (2 points). Find three piecewise smooth, positively oriented simple closed curves γ1, γ2, γ3 such that

for all j ∈ {1, 2, 3, 4} and all a ∈ Dj . Justify your answer.

Question 2 (2 points). For a ∈ D1, compute Rγfa(z) dz.

Question 3 (2 points). For a ∈ D2, compute Rγfa(z) dz.

Question 4 (2 points). For a ∈ D3, compute Rγfa(z) dz.

Question 5 (2 points). For a ∈ D4, compute Rγfa(z) dz.

Exercise 3 (5 points)

We say that an entire function f : C → C omits a set S ⊂ C if f(C) ⊂ C \ S.

Question 1 (1 point). Given w ∈ C and r ∈ R>0, define the function gw,r: C \ {w} → C by gw,r(z) = r/z−w . Prove that gw,r (C \ D (w, r )) ⊂ D (0, 1) \ {0} for all w ∈ C and all r ∈ R> 0.

Question 2 (2 points). Suppose that f : C → C is an entire function that omits some open disc D(w, r), with w ∈ C and r ∈ R>0. Prove that f is constant.

Question 3 (2 points). Suppose that f : C → C is an entire function that omits R≤0. Prove that f is constant.

Remark. In fact, Picard’s little theorem states that any nonconstant entire function omits at most one point.

Exercise 4 (5 points)

Suppose that f : D(0, 1) → C is a holomorphic function such that f  − n 1  = f  2/n  for all integers n ≥ 3.

Question 1 (2 points). Prove that f(−z) = f(2z) for all z ∈ D  0, 1 2  .

Question 2 (3 points). Prove that f is constant.

Exercise 5 (12 points)

The objective of the present exercise is to show that there is no entire function f : C → C such that f ◦ f = exp.

Question 1 (2 points). Suppose that u: C → C and v : C → C are two entire functions such that e u(z) = e v(z) for all z ∈ C. Prove that there is some k ∈ Z such that u(z) = v(z) + 2kπi for all z ∈ C.

From now on, to obtain a contradiction, assume that f : C → C is an entire function such that f (f(z)) = e z for all z ∈ C.

Question 2 (4 points). Prove that f has a holomorphic logarithm g : C → C – that is, show that there exists an entire function g : C → C such that f(z) = e g(z) for all z ∈ C.

Question 3 (2 points). Prove that there is some k ∈ Z such that g (f(z)) = z + 2kπi for all z ∈ C.

Question 4 (2 points). Prove that f : C → C is injective and g : C → C is surjective.

Question 5 (2 points). Conclude the proof that such a function f cannot exist.

Remark. In fact, there is no continuous function f : C → C such that f ◦ f = exp.