Math 5283, HW Set 4, Fall 2006
Due: Wednesday, October 4, 2006
1. Let P(z) = (z−a1)(z−a2)· · · (z−ak) and let γbe a closed curve with no
roots of Pon γ. Show that
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2πi Zγ
P0(z)
P(z)dz =n(γ, a1) + · · · +n(γ, ak).
2. Page 108, Problem 6 in Ahlfors.
3. Page 120, Problem 2 in Ahlfors.
4. Page 123, Problem 1 in Ahlfors.
5. Page 123, Problem 2 in Ahlfors.
6. Page 123, Problem 4 in Ahlfors.
7. Let f(z) be an analytic function on the unit disk |z| ≤ 1 and set
d:= sup
|z|≤1,|w|≤1
|f(z)−f(w)|.
Prove that f0(0) ≤d/2. (Hint: Start by showing
2f0(0) = 1
2πi ZC
f(ζ)−f(−ζ)
ζ2dζ,
where Cis any circle of radious r, 0 < r < 1 around the origin.)
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