3. Review of complex analysis
Definition 3.1. A domain D ⊆ C is an open connected subset of C.
Definition 3.2. For D ⊆ C a domain and f : D → C, we define the derivative of f at
z0 ∈ D to be
f (z) − f (z0 )
,
f 0 (z0 ) = lim
z→z0
z − z0
if it exists.
Definition 3.3. We call a function f : D → C holomorphic if it has a derivative at every
point in D.
Remark. If f is holomorphic on D, then not only is f infinitely differentiable in D, but
it equals its Taylor series expansion around every z0 ∈ D (in contrast to the situation
for functions of a real variable). The latter property is that of being analytic. In fact,
holomorphic and analytic are equivalent notions, and we will use the terms interchangeably.
Definition 3.4. We say that f : C → C is entire if it is holomorphic on C.
Theorem 3.5. If f is holomorphic on the closure of a domain D and ∂D is piecewise C 1 ,
then
Z
f (z) dz = 0.
∂D
There are two very useful theorems for proving that something is holomorphic:
R
Theorem 3.6 (Morera’s Theorem). If f is continuous on D, a domain, and R f = 0 for
any rectangle R ⊂ D with sides parallel to the coordinate axes, then f is holomorphic on D.
(Note that this is only a partial converse to the previous theorem, since, e.g., z1 is holomorphic on D = {z ∈ C : 1 < |z| < 2}, but the integral of z1 around a square inside D is not
0.)
Theorem 3.7 (Weierstrass). Let D ⊆ C be a domain and fn : D → C a sequence of
holomorphic functions on D. If fn converges to f uniformly on compact subsets of D, then
(m)
f : D → C is analytic on D (and fn → f (m) uniformly on compact subsets of D).
By fn → f uniformly on compacts, we mean that for every compact set K ⊂ D and every
ε > 0, there exists N ∈ N such that ∀n ≥ N ,
|fn (z) − f (z)| < ε,
∀z ∈ K.
Recall also that a set is compact if and only if it is closed and bounded (Heine–Borel). By
Cauchy’s criterion for convergence, one has that fn → f uniformly on K ⇔ for all ε > 0,
∃N ∈ N such that ∀n, m ≥ N ,
|fn (z) − fm (z)| < ε ∀z ∈ K.
As an application, let us see that ζ(s) is analytic on the domain that we have considered
so far:
Lemma 3.8. The Riemann zeta-function is holomorphic on D = {s ∈ C : <(s) > 1}.
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Proof. Let
fN (s) =
N
X
n−s .
n=1
Then fN is entire, since it is a finite sum of the holomorphic functions n−s = exp(−s log n).
Let K ⊂ D be a compact set. Then there exists δ = δ(K) > 0 such that K ⊂ {s ∈ C :
<(s) ≥ 1 + δ}. Hence, for all s ∈ K we have that
∞
∞
∞
X 1
X
X
1
1
|ζ(s) − fN (s)| ≤ ≤
≤
s
<(s)
1+δ
n n=N +1 n
n
n=N +1
n=N +1
Z ∞
dt
1
<
=
→ 0 as N → ∞.
1+δ
δN δ
N t
Since the rate of convergence to 0 depends only on K (via δ) and not on s ∈ K, the
convergence fN (s) → ζ(s) is uniform on K. Hence by Weierstrass, ζ(s) is analytic on D. Singularities.
Definition 3.9. We say f : D → C has an isolated singularity at z0 ∈ D if f is analytic on
a punctured neighbourhood of z0 ,
{z ∈ C : 0 < |z − z0 | < r},
for some r > 0. An isolated singularity at z0 implies that there is a convergent Laurent series
expansion
∞
X
f (z) =
an (z − z0 )n
n=−∞
for z in this punctured neighbourhood.
There are three types of singularities:
(1) Removable singularities, where an = 0, ∀n < 0, e.g.
sin z
z2 z4
=1−
+
− ...
z
3!
5!
has a removable singularity at z = 0;
(2) Pole of order m ≥ 1, if an = 0 ∀n < −m, and a−m 6= 0, e.g.
pole of order 1) at z = 0;
(3) Essential singularity, if it is not of the other two types.
1
z
has a simple pole (=
Definition 3.10. The residue of f at an isolated singularity z0 is a−1 , the coefficient of
(z − z0 )−1 in the Laurent expansion, and it is denoted Resz=z0 f (z).
One can show that
Resz=z0 f (z) = lim (z − z0 )f (z)
z→z0
if f has a simple pole at z = z0 , and in general, for a pole of order m,
1
dm−1
m
Resz=z0 f (z) = lim
(z
−
z
)
f
(z)
.
0
z→z0 (m − 1)! dz m−1
2
Theorem 3.11 (Residue Theorem). Let D ⊂ C be a bounded domain with a piecewise
C 1 boundary, and let f : D → C be analytic on D except at a finite number of points
z1 , . . . , zn ∈ D. Then
Z
n
X
1
f (z) dz =
Resz=zj f (z).
2πi ∂D
j=1
Theorem 3.12 (Uniqueness Principle). Let f, g be analytic on D, a domain, and suppose
that f (z) = g(z) for all z in a subset of D with a limit point in D. Then f = g on D.
Corollary 3.13 (Principle of Analytic Continuation). If f : D → C is holomorphic and
D ⊆ D0 , then there is at most one extension F : D0 → C such that F D = f and F is
holomorphic on D0 . Such an F is called the analytic continuation of f to D0 .
c
Andrew
R. Booker, Tim D. Browning, Min Lee and Dave Platt, University of Bristol 2017.
This material is copyright of Andrew R. Booker, Tim D. Browning, Min Lee and Dave Platt
at the University of Bristol unless explicitly stated otherwise. It is provided exclusively for
educational purposes at the University of Bristol, and is to be downloaded or copied for your
private study only.
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