Math 411: Honours Complex Variables - University of Alberta

44 CHAPTER 6. CONVERGENCE OF HOLOMORPHIC FUNCTIONS
Proof. If γ : [a,b] → C is a curve (and thus continuous) then {γ} = γ([a,b]) is
compact. Hence, Lemma 6.1 applies.
Theorem 6.2 (Weierstraß Theorem). Let D ⊂ C be open, let f1,f2,... : D → C
converges to f : D → C compactly. Then f is
be holomorphic such that (fn) ∞ n=1
holomorphic, and (f (k)
n ) ∞ n=1 converges compactly to f (k) for each k ∈ N.
Proof. By Theorem 6.1, f is continuous.
�
To see that f is holomorphic, let ∆ ⊂ D be a triangle. By Goursat’s Lemma,
∂∆fn(ζ)dζ = 0 holds for all n ∈ N. From Lemma 6.2, we conclude that
� �
f(ζ)dζ = lim fn(ζ)dζ = 0,
n→∞
∂∆
i.e. f satisfies the Morera condition and thus is holomorphic.
Let z0 ∈ D, and let 0 < r < R be such that Br[z0] ⊂ BR(z0) ⊂ BR[z0] ⊂ D. For
any z ∈ Br(z0), we have
|f ′ n (z)−f′ (z)| = 1
2π
Let ǫ > 0, and choose N ∈ N such that
∂∆
��
�
�
fn(ζ)−f(ζ)
�
∂BR(z0) (ζ −z) 2
�
�
dζ�
�
≤ 1
2π ℓ(∂BR(z0))
�
�
sup �
fn(ζ)−f(ζ)
� (ζ −z) 2
�
�
�
�
≤
ζ∈∂BR(z0)
R
(R−r) 2
sup |fn(ζ)−f(ζ)|.
ζ∈∂BR(z0)
|fn(ζ)−f(ζ)|< ǫ (R−r)2
R
for all n ≥ N and ζ ∈ ∂BR(z0). Then it follows from the above estimates that
|f ′ n (z)−f′ (z)|≤ ǫforall n ≥ N andz ∈ Br(z0). Consequently, (f ′ n |Br(z0)) ∞ n=1 converges
to f ′ |Br(z0) uniformly on Br(z0). As z0 ∈ D is arbitrary, this means that (f ′ n) ∞ n=1
converges to f locally uniformly, i.e. compactly, on D.
For higher derivatives, the claim now follows by induction.
Lemma 6.3. Let z0 ∈ C, let r > 0, and let z ∈ Br(z0). Then
1
ζ −z =
∞�
n=0
converges absolutely and uniformly on ∂Br(z0).
1 n
(ζ −z0) n+1(z −z0)