Math 411: Honours Complex Variables - University of Alberta

Corollary 11.2.1. Let D be an open, connected subset of C. Then D is simply
connected ⇐⇒ every holomorphic function on D has an antiderivative.
Problem 11.1. Let D ⊂ C be open and connected such that, for each holomorphic
f : D → C, there is a sequence (pn) ∞ n=1 of polynomials converging to f compactly
on D. Show that D is simply connected.
Definition. Let D ⊂ C be open and connected and n ∈ N. We say that D admits
(a) holomorphic logarithms if, for every holomorphic function f : D → C with
Z(f) = ∅, there exists a holomorphic function g: D → C with f = exp◦g;
(b) holomorphicnth rootsifforeveryholomorphicfunctionf: D → CwithZ(f) = ∅,
there exists a holomorphic function hn: D → C with f(z) = [hn(z)] n for z ∈ D;
(c) holomorphic roots if D admits holomorphic nth roots for each n ∈ N.
Corollary 11.2.2 (Holomorphic Logarithms). A simply connected domain admits
holomorphic logarithms.
Proof. This follows from Corollary 11.2.1 and Problem 5.1(a).
Corollary 11.2.3 (Holomorphic Roots). A simply connected domain admits holomorphic
roots.
�
for n ∈ N.
Proof. Let g be such that f = exp◦g, and set hn := exp◦ � g
n
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