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