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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Definition. The length <strong>of</strong> a piecewise smooth curve γ: [a,b] → C is defined as<br />
ℓ(γ) :=<br />
n�<br />
� aj<br />
|γ ′ (t)|dt,<br />
j=1<br />
where a = a0 < a1 < ··· < an = b is a partition such that γ|[aj−1,aj] is continuously<br />
differentiable for j = 1,...,n.<br />
Definition. Let γ: [a,b] → C be a piecewise smooth curve, let a = a0 < a1 < ··· <<br />
an = b beapartitionsuch that γ|[aj−1,aj] iscontinuously differentiable forj = 1,...,n,<br />
and let f : {γ} → C be continuous. Then the line integral (or contour integral) <strong>of</strong> f<br />
along γ is defined as<br />
�<br />
γ<br />
�<br />
f :=<br />
γ<br />
f(ζ)dζ =<br />
aj−1<br />
n�<br />
� aj<br />
j=1<br />
aj−1<br />
f(γ(t))γ ′ (t)dt.<br />
Properties <strong>of</strong> the Line Integral. 1. Let γ be a piecewise smooth curve, let<br />
λ,µ ∈ C, and let f,g: {γ} → C be continuous. Then we have<br />
� � �<br />
(λf +µg) = λ f +µ g.<br />
γ<br />
2. Let γ be a piecewise smooth curve, let f : {γ} → C be continuous, and let<br />
C ≥ 0 be such that |f(ζ)|≤ C for ζ ∈ {γ}. Then<br />
��<br />
�<br />
� �<br />
�<br />
� f�<br />
� ≤ Cℓ(γ)<br />
holds.<br />
γ<br />
3. Let γ : [c,d] → C be a piecewise smooth curve, let φ : [a,b] → [c,d] be a<br />
continuously differentiable function with φ(a) = c and φ(b) = d, and let f :<br />
{γ} → C be continuous. Then we have<br />
� �<br />
f = f.<br />
γ<br />
4. Let D ⊂ C be open, and let f : D → C be continuous with antiderivative<br />
F : D → C; i.e. F is complex differentiable at each z ∈ D, with F ′ (z) = f(z).<br />
Then �<br />
f = F(γ(b))−F(γ(a))<br />
γ<br />
γ◦φ<br />
holds for every piecewise smooth curve γ: [a,b] → D.<br />
γ<br />
γ<br />
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