Math 411: Honours Complex Variables - University of Alberta

10 CHAPTER 2. COMPLEX DIFFERENTIATION
Proof. As over R.
Proposition 2.3. Let D,E ⊂ C, let g : D → C and f : E → C be such that
g(D) ⊂ E, and let z0 ∈ intD be such that w0 := g(z0) ∈ intE. Further, suppose
that g is complex differentiable at z0 and f is complex differentiable at w0. Then f◦g
is complex differentiable at z0 with
Proof. As over R.
Examples.
(f ◦g) ′ (z0) = f ′ (g(z0))g ′ (z0).
1. All constant functions are (on all of C) complex differentiable, as is z ↦→ z on
C. Consequently, all complex polynomials are complex differentiable on all of
C, and rational functions are complex differentiable wherever they are defined.
2. Let
f: C → C, z ↦→ ¯z,
and let z0 = x0+iy0 ∈ C. Assume that f is complex differentiable at z0. Then
we have
as well as
f ′ ¯z − ¯z0
(z0) = lim
z→z0
z −z0
(x0 −iy)−(x0 −iy0)
= lim
y→y0 (x0 +iy)−(x0 +iy0)
i(y0 −y)
= lim
y→y0 i(y −y0)
= −1
f ′ ¯z − ¯z0
(z0) = lim
z→z0
z −z0
(x−iy0)−(x0 −iy0)
= lim
x→x0 (x+iy0)−(x0 +iy0)
x−x0
= lim
x→x0 x−x0
= 1,
which is impossible. Hence, f is not complex differentiable at any z0 ∈ C. (On
the other hand, f is continuously partially differentiable—as a function of two
real variables—on all of C.)
Lemma 2.1. The following are equivalent for an R-linear map T : C → C: