Math 411: Honours Complex Variables - University of Alberta

Definition. The length of a piecewise smooth curve γ: [a,b] → C is defined as
ℓ(γ) :=
n�
� aj
|γ ′ (t)|dt,
j=1
where a = a0 < a1 < ··· < an = b is a partition such that γ|[aj−1,aj] is continuously
differentiable for j = 1,...,n.
Definition. Let γ: [a,b] → C be a piecewise smooth curve, let a = a0 < a1 < ··· <
an = b beapartitionsuch that γ|[aj−1,aj] iscontinuously differentiable forj = 1,...,n,
and let f : {γ} → C be continuous. Then the line integral (or contour integral) of f
along γ is defined as
�
γ
�
f :=
γ
f(ζ)dζ =
aj−1
n�
� aj
j=1
aj−1
f(γ(t))γ ′ (t)dt.
Properties of the Line Integral. 1. Let γ be a piecewise smooth curve, let
λ,µ ∈ C, and let f,g: {γ} → C be continuous. Then we have
� � �
(λf +µg) = λ f +µ g.
γ
2. Let γ be a piecewise smooth curve, let f : {γ} → C be continuous, and let
C ≥ 0 be such that |f(ζ)|≤ C for ζ ∈ {γ}. Then
��
�
� �
�
� f�
� ≤ Cℓ(γ)
holds.
γ
3. Let γ : [c,d] → C be a piecewise smooth curve, let φ : [a,b] → [c,d] be a
continuously differentiable function with φ(a) = c and φ(b) = d, and let f :
{γ} → C be continuous. Then we have
� �
f = f.
γ
4. Let D ⊂ C be open, and let f : D → C be continuous with antiderivative
F : D → C; i.e. F is complex differentiable at each z ∈ D, with F ′ (z) = f(z).
Then �
f = F(γ(b))−F(γ(a))
γ
γ◦φ
holds for every piecewise smooth curve γ: [a,b] → D.
γ
γ
23