Math 411: Honours Complex Variables - University of Alberta

Chapter 2
Complex Differentiation
Definition. Let D ⊂ C, and let z0 be an interior point of D, i.e. there exists ǫ > 0
such that Bǫ(z0) := {z ∈ C : |z − z0|< ǫ} ⊂ D. A function f : D → C is called
complex differentiable at z0 if
exists.
f ′ f(z)−f(z0)
(z0) := lim
z→z0 z −z0
Proposition 2.1. Let D ⊂ C, let z0 ∈ intD, and let f: D → C be complex differentiable
at z0. Then f is continuous at z0.
f(z)−f(z0)
Proof. Since lim
z→z0 z −z0
f(z)−f(z0)
0 = lim(z
−z0) lim
z→z0 z→z0 z −z0
so that f(z0) = lim f(z).
z→z0
exists, we have
= lim(z
−z0)
z→z0
f(z)−f(z0)
z −z0
= lim(f(z)−f(z0)),
z→z0
Proposition 2.2. Let D ⊂ C, and let f,g : D → C be complex differentiable at
z0 ∈ intD. Then the following functions are complex differentiable at z0: f +g, fg,
and, if g(z0) �= 0, f
g
and
. Moreover, we have:
(f +g) ′ (z0) = f ′ (z0)+g ′ (z0),
(fg) ′ (z0) = f ′ (z0)g(z0)+f(z0)g ′ (z0),
� � ′
f
g
(z0) = f′ (z0)g(z0)−f(z0)g ′ (z0)
g(z0) 2
.
9