Math 411: Honours Complex Variables - University of Alberta

40 CHAPTER 5. CAUCHY’S INTEGRAL THEOREM AND FORMULA
(iv) for each z0 ∈ D, there exists r > 0 with Br[z0] ⊂ D and
f(z) = 1
�
f(ζ)
2πi ζ −z dζ
for z ∈ Br(z0);
∂Br(z0)
(v) f is infinitely often complex differentiable on D;
(vi) for each z0 ∈ D, there exists a neighbourhood U ⊂ D of z0 such that f has an
antiderivative on U.
Proof. (i) =⇒ (ii) is Goursat’s Lemma.
(i) =⇒ (iii) is the Cauchy Integral Formula for circles, and (iii) =⇒ (iv) is trivial.
(iv) =⇒(v) followsimmediately fromTheorem 5.4, and(v) =⇒ (i) isagaintrivial.
(ii) =⇒ (vi) follows from Theorem 5.2 because every z0 ∈ D has an open, starshaped
neighbourhood contained in D.
(vi) =⇒ (v): Let z0 ∈ D, and let U ⊂ D be a neighbourhood of z0 such that f
has an antiderivative, say F, on U. Then F is holomorphic on U. Applying (i) =⇒
(v) to F, we see that F is infinitely often complex differentiable on U. Consequently,
f = F ′ is infinitely complex differentiable on U. Since z0 ∈ D was arbitrary, we
conclude that f is infinitely often complex differentiable on D.
We conclude this chapter with Liouville’s Theorem and its application to the
Fundamental Theorem of Algebra.
Definition. A holomorphic function defined on all of C is called entire.
Theorem 5.6 (Liouville’s Theorem). Let f : C → C be a bounded entire function.
Then f is constant.
Proof. We will show that f ′ ≡ 0.
Let C ≥ 0 be such that |f(z)|≤ C for all z ∈ C. Let z ∈ C be arbitrary, and let
r > 0. By the generalized Cauchy integral formula, we have
|f ′ (z)|= 1
2π
��
�
�
�
∂Br(z)
f(ζ)
(ζ −z)
2 dζ
�
�
�
�
1
≤
2π ℓ(∂Br(z))
|f(ζ)| 1
sup ≤
ζ∈∂Br(z) |ζ −z| 2 2π 2πrC
C
=
r2 r .
Letting r → ∞, we obtain f ′ (z) = 0. This completes the proof.
Corollary 5.6.1 (Fundamental Theorem of Algebra). Let p be a non-constant polynomial
with complex coefficients. Then p has a zero.
Proof. Assume that p has no zero. Then the function
f: C → C, z ↦→ 1
p(z)