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