Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
24 CHAPTER 4. COMPLEX LINE INTEGRALS<br />
Definition. A curve γ: [a,b] → C is called closed if γ(a) = γ(b).<br />
Proposition 4.1. Let D ⊂ C be open, and let f : D → C be continuous with an<br />
antiderivative. Then �<br />
f = 0 holds for each closed, piecewise smooth curve γ in D.<br />
Example. Let z0 ∈ C, let r > 0, and let<br />
γ<br />
γ: [0,2π] → C, θ ↦→ re iθ +z0,<br />
i.e. γ is a counterclockwise-oriented circle centered at z0 with radius r.<br />
Let n ∈ Z, and consider �<br />
γ (ζ −z0) ndζ. For n �= −1, let<br />
n+1 (z −z0)<br />
F : C → C, z ↦→ ,<br />
n+1<br />
so that F ′ (z) = (z −z0) n for all z ∈ C. It follows that �<br />
On the other hand, we have<br />
Consequently,<br />
�<br />
has no antiderivative.<br />
γ<br />
(ζ −z0) −1 dζ =<br />
� 2π<br />
0<br />
γ (ζ −z0) n dζ = 0.<br />
rieiθ � 2π<br />
dθ = idt = 2πi.<br />
reiθ 0<br />
C\{z0} → C, z ↦→ 1<br />
z −z0<br />
Recall the following definition from multivariable calculus:<br />
Definition. A subset D ⊂ C is called connected if there are no open sets U,V ⊂ C<br />
with<br />
• U ∩D �= ∅ �= V ∩D;<br />
• U ∪V ⊃ D;<br />
• U ∩V ⊂ C\D.<br />
In other words, there are no open sets U and V, each containing points <strong>of</strong> D, such<br />
that every point <strong>of</strong> D lies in exactly one <strong>of</strong> the sets U and V.<br />
Definition. Let D ⊂ C be open. A function f : D → C is called locally constant if,<br />
for each z0 ∈ D, there exists ǫ > 0 such that Bǫ(z0) ⊂ D and f is constant on Bǫ(z0).<br />
The following curve constructions will be useful in understanding the relation<br />
between locally constant functions and connectivity.