3. I> Let Ω ⊆ C be open. We say that Ω is simply connected provided that Ω is connected and Indγ z = 0 for every closed contour γ in Ω and every z ∈ C \ Ω. (Loosely, this means Ω has no “holes.” This...

3. I> Let Ω
⊆
C be open. We say that Ω is
simply connected
provided that Ω is connected and Indγ

z
= 0 for every closed contour
γ
in Ω and every
z

∈
C
\
Ω. (Loosely, this

means Ω has no “holes.” This is not the traditional definition.)

(a) Suppose that Ω
⊆
C is a simply connected domain. Show that

r

f
(z)
dz
=

C1

r

f
(z)
dz

C2

for any
f

∈

H(Ω) if
C1, C2
⊆
Ω are contours with the same respective initial and terminal points.

(b) Show that if Ω
⊆
C is a simply connected domain and
f

∈

H(Ω), then
f
has an antiderivative
F

∈

H(Ω). (Compare to Theorem 3.1.3.)