Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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1. Given a < b < c and two curves γ1 : [a,b] → C and γ2 : [b,c] → C with<br />
γ1(b) = γ2(b), the concatenation <strong>of</strong> γ1 and γ2 is the curve<br />
�<br />
γ1(t), t ∈ [a,b],<br />
γ1 ⊕γ2: [a,c] → C, t ↦→<br />
γ2(t), t ∈ [b,c].<br />
If γ1 and γ2 are piecewise smooth, then so is γ1 ⊕γ2, and we have<br />
�<br />
γ1⊕γ2<br />
�<br />
f =<br />
for each continuous f: {γ1}∪{γ2} → C.<br />
γ1<br />
�<br />
f +<br />
2. For any curve γ: [a,b] → C, the reversed curve is defined as<br />
γ − : [a,b] → C, t ↦→ γ(a+b−t).<br />
If γ is piecewise smooth, then so is γ− , and we have<br />
� �<br />
f = − f<br />
for each continuous f: {γ} → C.<br />
γ −<br />
3. We denote the straight line segment {z0 +t(z −z0) : t ∈ [0,1]} by [z0,z].<br />
Proposition 4.2 (Locally Constant vs. Connectivity). Let D ⊂ C be open. Then<br />
the following are equivalent:<br />
(i) D is connected;<br />
(ii) every locally constant function f: D → C is constant;<br />
(iii) for any z,w ∈ D, there exists a piecewise smooth curve γ: [a,b] → D such that<br />
γ(a) = z and γ(b) = w.<br />
Pro<strong>of</strong>. (iii) =⇒ (ii): Let f: D → C be a locally constant function, and let z,w ∈ D.<br />
Let γ: [a,b] → D be a piecewise smooth curve with γ(a) = z and γ(b) = w. Since f<br />
is locally constant, the function<br />
γ<br />
γ2<br />
[a,b] → C, t ↦→ f(γ(t))<br />
is differentiable with zero derivative and therefore constant. It follows that f(z) =<br />
f(γ(a)) = f(γ(b)) = f(w).<br />
(ii) =⇒ (i): Suppose that D is not connected. Then there exist non-empty open<br />
sets U,V ⊂ C with U ∩V = ∅ and U ∪V = D. Define<br />
f: D → C, z ↦→<br />
f<br />
� 0, z ∈ U,<br />
1, z ∈ V.<br />
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