Math 411: Honours Complex Variables - University of Alberta

Proof. (i) =⇒ (ii) is Corollary 11.2.2, (ii) =⇒ (iii) is shown in the proof of Corollary
11.2.3, (iii) =⇒ (iv) is trivial, (iv) =⇒ (v) follows from Theorem 17.3, and (v)
=⇒ (i) is implied by Proposition 17.1.
(i) ⇐⇒ (vi) is Corollary 11.2.1 and (vi) ⇐⇒ (vii) follows from Theorem 4.1.
(i) =⇒ (viii) follows from Theorem 11.1, and (viii) =⇒ (vii) is established in the
proof of Theorem 11.2.
(v) =⇒ (ix): Let u: D → R be harmonic. If D = C, the existence of a harmonic
conjugate is immediate by Theorem 14.1. So suppose that D �= C. Hence, there is a
biholomorphic map f: D → D. It is easily seen that ũ := u◦f −1 : D → R is harmonic
and by Theorem 14.1 has a harmonic conjugate ˜v: D → R. Then v := ˜v ◦f: D → R
is a harmonic conjugate of u.
(ix) =⇒(ii): Letf: D → C beholomorphicsuch thatZ(f) = ∅. Thenu := log|f|
is harmonic and thus has a harmonic conjugate v : D → R so that g := u + iv is
holomorphic. On D we have
so that
|expg|= |exp(u+iv)|= expu = |f|,
D → C, z ↦→ f(z)
exp(g(z))
is a holomorphic function whose range lies on ∂D and therefore isn’t open. By the
Open Mapping Theorem, this means that there exists c ∈ ∂D such that f(z) =
cexp(g(z)) for z ∈ D. Choose θ ∈ R with exp(iθ) = c, and note that f(z) =
exp(g(z)+iθ) for z ∈ D.
Definition. Two (not necessarily piecewise smooth) closed curves γ1,γ2: [0,1] → D
with γ1(0) = γ2(0) andγ1(1) = γ2(1) arecalled path homotopicifthere is acontinuous
function Γ: [0,1]×[0,1] → D such that,
for t ∈ [0,1] and
for all s ∈ [0,1].
Γ(0,t) = γ1(t) and Γ(1,t) = γ2(t)
Γ(s,0) = γ1(0) and Γ(s,1) = γ1(1)
Definition. A closed curve γ is called homotopic to zero if γ and the constant curve
γ(0) are path homotopic.
Further Characterizations of Simply Connected Domains. There are further
conditions that characterize simply connected domains. We will only state them,
without giving proofs. Simple connectedness is also equivalent to:
(x) every (not necessarily smooth) curve in D is homotopic to zero;
(xi) D is homeomorphic to D.
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