lOMoARcPSD|11409177
Complex Analysis (Lecture notes, updated November 22nd)
Linear algebra 1 (University of Oxford)
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A2: COMPLEX ANALYSIS
These are notes for the later two-thirds of the course A2: Metric Spaces
and Complex Analysis, covering the material on complex analysis. They
are closely based on a previous version of the notes by Kevin McGerty.
Contact Ben Green (Sections 1–3) or Panos Papasoglu (the rest of the course)
if you have any comments or corrections.
Synopsis
Basic geometry and topology of the complex plane, including the equa-
tions of lines and circles. Extended complex plane, Riemann sphere, stere-
ographic projection. Möbius transformations acting on the extended com-
plex plane. Möbius transformations take circlines to circlines. [3]
Complex differentiation. Holomorphic functions. Cauchy-Riemann equa-
tions (including z, z version). Real and imaginary parts of a holomorphic
function are harmonic. [2]
Recap on power series and differentiation of power series. Exponen-
tial function and logarithm function. Fractional powers–examples of mul-
tifunctions. The use of cuts as method of defining a branch of a multifunc-
tion. [3]
Path integration. Cauchy’s Theorem. (Sketch of proof only–students re-
ferred to various texts for proof.) Fundamental Theorem of Calculus in the
path integral/holomorphic situation. [2]
Cauchy’s Integral formulae. Taylor expansion. Liouville’s Theorem.
Identity Theorem. Morera’s Theorem. [4]
Laurent’s expansion. Classification of isolated singularities. Calculation
of principal parts, particularly residues. [2]
Residue Theorem. Evaluation of integrals by the method of residues
(straightforward examples only but to include the use of Jordan’s Lemma
and simple poles on contour of integration). [3]
Conformal mappings. Riemann mapping theorem (no proof): Möbius
transformations, exponential functions, fractional powers; mapping regions
(not Christoffel transformations or Joukowski’s transformation). [3]
Date: November 22, 2020.
1
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, lOMoARcPSD|11409177
2 A2: COMPLEX ANALYSIS
1. G EOMETRY AND TOPOLOGY OF THE COMPLEX PLANE
The aim of this part of the course is to study functions f : C → C, asking
what it means for them to be differentiable, how to integrate them, and
looking at the applications of all this. Before we begin, we record some
basic properties of the complex numbers C, much of which is revision from
Prelims.
1.1. C as a metric space. We can identify C with the plane R2 by taking
real and imaginary parts. Thus we have mutually inverse bijections
z 7→ (ℜz, ℑz)
from C to R2 , and
(x, y) 7→ x + iy
from R2 to C. As we saw in the first part of the course, R2 is a metric space
with the metric induced from the Euclidean norm
p
k(x, y)k2 = x2 + y 2 .
This gives a metric on C by the identification C ∼
= R2 described above.
If z = ℜz + iℑz is a complex number we write |z| (called the modulus) for
this Euclidean norm, that is,
p
|z| = (ℜz)2 + (ℑz)2 .
The distance between two points z, w ∈ C is then |z − w|.
Let us write down some basic properties of the modulus |z|. Recall that
eiθ = cos θ + i sin θ when θ ∈ R. For now, we will take this as the definition
of eiθ , which is more-or-less what was done in Prelims Complex Analysis.
Later on we will define the complex exponential function ez and link the
two concepts.
Lemma 1.1. Let z, w ∈ C. Then
(1) |z|2 = z z̄, where z̄ is the complex conjugate of z;
(2) If z = reiθ , where r ∈ [0, ∞) and θ ∈ R, then |z| = r;
(3) |zw| = |z||w|.
Proof. (1) If z = a + ib then z z̄ = (a + ib)(a − ib) = a2 + b2 .
(2) We have z = r cos θ + ir sin θ and so
p
|z| = r2 cos2 θ + r2 sin2 θ = r.
(3) One can calculate directly, writing z = a + ib and w = c + id. Alterna-
tively, write z = reiθ , w = r′ eiα , and then observe that zw = rr′ ei(θ+α) and
use (2).
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, lOMoARcPSD|11409177
A2: COMPLEX ANALYSIS 3
1.2. Topological properties of C. One can make sense of the notion of
open set, closure, interior and so on by identifying C with R2 .
In the language of this part of the course, U ⊂ C being open means that
if z ∈ U then some ball B(z, ε), ε > 0, also lies in U , where
B(z, ε) := {w ∈ C : |z − w| < ε}.
In complex analysis it is often convenient to work with connected open
sets, and these are called domains.
Definition 1.2. A connected open subset D ⊆ C of the complex plane will
be called a domain.
We saw in the metric spaces part of the course (Chapter 7) that, for open
subsets of normed spaces (such as R2 with the Euclidean metric), the no-
tions of connectedness and path-connectedness are the same thing. There-
fore domains are always path-connected.
1.3. Geometry of C. Let us take a closer look at the geometry of the com-
plex plane in terms of the distance |z − w|. When we talk about lines and
circles in C, we mean sets that are lines and circles in R2 (under the identi-
fication of R2 with C).
Lemma 1.3 (Lines). Let a, b ∈ C be distinct complex numbers. Then the set
{z ∈ C : |z − a| = |z − b|} is a line. Conversely, every line can be written in this
form.
Proof. Given a and b, the set of z such that |z − a| = |z − b| is the set of
points equidistant from a and b, which is the perpendicular bisector of the
line segment ab. Conversely, every line is the perpendicular bisector of
some line segment.
Remarks. Sometimes, the set of all complex numbers satisfying some
given equation is called a locus. Thus the locus of complex numbers sat-
isfying |z − a| = |z − b| is a line. The representation of lines in the above
form is very much non-unique: for example, the x-axis (the set of z with
zero imaginery part) can be described as {z : |z − a| = |z − ā|} for any
complex number a.
Now we turn to circles. Evidently, the set {z ∈ C : |z − c| = r}, where
c ∈ C and r ∈ (0, ∞), is a circle centred on c and with radius r. Conversely,
every circle can be written in this form. Less obvious is the following.
Lemma 1.4. Let a, b ∈ C be distinct complex numbers, and let λ ∈ (0, ∞),
λ 6= 1. Then the locus of complex numbers satisfying |z − a| = λ|z − b| is a circle.
Conversely, every circle can be written in this form.
Proof. Without loss of generality, b = 0 (a translate of a circle is a circle).
Now observe the identity
|tz + a|2 = t(t + 1)|z|2 − t|z − a|2 + (t + 1)|a|2 ,
Downloaded by Ashley Green ()
Complex Analysis (Lecture notes, updated November 22nd)
Linear algebra 1 (University of Oxford)
StuDocu is not sponsored or endorsed by any college or university
Downloaded by Ashley Green ()
, lOMoARcPSD|11409177
A2: COMPLEX ANALYSIS
These are notes for the later two-thirds of the course A2: Metric Spaces
and Complex Analysis, covering the material on complex analysis. They
are closely based on a previous version of the notes by Kevin McGerty.
Contact Ben Green (Sections 1–3) or Panos Papasoglu (the rest of the course)
if you have any comments or corrections.
Synopsis
Basic geometry and topology of the complex plane, including the equa-
tions of lines and circles. Extended complex plane, Riemann sphere, stere-
ographic projection. Möbius transformations acting on the extended com-
plex plane. Möbius transformations take circlines to circlines. [3]
Complex differentiation. Holomorphic functions. Cauchy-Riemann equa-
tions (including z, z version). Real and imaginary parts of a holomorphic
function are harmonic. [2]
Recap on power series and differentiation of power series. Exponen-
tial function and logarithm function. Fractional powers–examples of mul-
tifunctions. The use of cuts as method of defining a branch of a multifunc-
tion. [3]
Path integration. Cauchy’s Theorem. (Sketch of proof only–students re-
ferred to various texts for proof.) Fundamental Theorem of Calculus in the
path integral/holomorphic situation. [2]
Cauchy’s Integral formulae. Taylor expansion. Liouville’s Theorem.
Identity Theorem. Morera’s Theorem. [4]
Laurent’s expansion. Classification of isolated singularities. Calculation
of principal parts, particularly residues. [2]
Residue Theorem. Evaluation of integrals by the method of residues
(straightforward examples only but to include the use of Jordan’s Lemma
and simple poles on contour of integration). [3]
Conformal mappings. Riemann mapping theorem (no proof): Möbius
transformations, exponential functions, fractional powers; mapping regions
(not Christoffel transformations or Joukowski’s transformation). [3]
Date: November 22, 2020.
1
Downloaded by Ashley Green ()
, lOMoARcPSD|11409177
2 A2: COMPLEX ANALYSIS
1. G EOMETRY AND TOPOLOGY OF THE COMPLEX PLANE
The aim of this part of the course is to study functions f : C → C, asking
what it means for them to be differentiable, how to integrate them, and
looking at the applications of all this. Before we begin, we record some
basic properties of the complex numbers C, much of which is revision from
Prelims.
1.1. C as a metric space. We can identify C with the plane R2 by taking
real and imaginary parts. Thus we have mutually inverse bijections
z 7→ (ℜz, ℑz)
from C to R2 , and
(x, y) 7→ x + iy
from R2 to C. As we saw in the first part of the course, R2 is a metric space
with the metric induced from the Euclidean norm
p
k(x, y)k2 = x2 + y 2 .
This gives a metric on C by the identification C ∼
= R2 described above.
If z = ℜz + iℑz is a complex number we write |z| (called the modulus) for
this Euclidean norm, that is,
p
|z| = (ℜz)2 + (ℑz)2 .
The distance between two points z, w ∈ C is then |z − w|.
Let us write down some basic properties of the modulus |z|. Recall that
eiθ = cos θ + i sin θ when θ ∈ R. For now, we will take this as the definition
of eiθ , which is more-or-less what was done in Prelims Complex Analysis.
Later on we will define the complex exponential function ez and link the
two concepts.
Lemma 1.1. Let z, w ∈ C. Then
(1) |z|2 = z z̄, where z̄ is the complex conjugate of z;
(2) If z = reiθ , where r ∈ [0, ∞) and θ ∈ R, then |z| = r;
(3) |zw| = |z||w|.
Proof. (1) If z = a + ib then z z̄ = (a + ib)(a − ib) = a2 + b2 .
(2) We have z = r cos θ + ir sin θ and so
p
|z| = r2 cos2 θ + r2 sin2 θ = r.
(3) One can calculate directly, writing z = a + ib and w = c + id. Alterna-
tively, write z = reiθ , w = r′ eiα , and then observe that zw = rr′ ei(θ+α) and
use (2).
Downloaded by Ashley Green ()
, lOMoARcPSD|11409177
A2: COMPLEX ANALYSIS 3
1.2. Topological properties of C. One can make sense of the notion of
open set, closure, interior and so on by identifying C with R2 .
In the language of this part of the course, U ⊂ C being open means that
if z ∈ U then some ball B(z, ε), ε > 0, also lies in U , where
B(z, ε) := {w ∈ C : |z − w| < ε}.
In complex analysis it is often convenient to work with connected open
sets, and these are called domains.
Definition 1.2. A connected open subset D ⊆ C of the complex plane will
be called a domain.
We saw in the metric spaces part of the course (Chapter 7) that, for open
subsets of normed spaces (such as R2 with the Euclidean metric), the no-
tions of connectedness and path-connectedness are the same thing. There-
fore domains are always path-connected.
1.3. Geometry of C. Let us take a closer look at the geometry of the com-
plex plane in terms of the distance |z − w|. When we talk about lines and
circles in C, we mean sets that are lines and circles in R2 (under the identi-
fication of R2 with C).
Lemma 1.3 (Lines). Let a, b ∈ C be distinct complex numbers. Then the set
{z ∈ C : |z − a| = |z − b|} is a line. Conversely, every line can be written in this
form.
Proof. Given a and b, the set of z such that |z − a| = |z − b| is the set of
points equidistant from a and b, which is the perpendicular bisector of the
line segment ab. Conversely, every line is the perpendicular bisector of
some line segment.
Remarks. Sometimes, the set of all complex numbers satisfying some
given equation is called a locus. Thus the locus of complex numbers sat-
isfying |z − a| = |z − b| is a line. The representation of lines in the above
form is very much non-unique: for example, the x-axis (the set of z with
zero imaginery part) can be described as {z : |z − a| = |z − ā|} for any
complex number a.
Now we turn to circles. Evidently, the set {z ∈ C : |z − c| = r}, where
c ∈ C and r ∈ (0, ∞), is a circle centred on c and with radius r. Conversely,
every circle can be written in this form. Less obvious is the following.
Lemma 1.4. Let a, b ∈ C be distinct complex numbers, and let λ ∈ (0, ∞),
λ 6= 1. Then the locus of complex numbers satisfying |z − a| = λ|z − b| is a circle.
Conversely, every circle can be written in this form.
Proof. Without loss of generality, b = 0 (a translate of a circle is a circle).
Now observe the identity
|tz + a|2 = t(t + 1)|z|2 − t|z − a|2 + (t + 1)|a|2 ,
Downloaded by Ashley Green ()
