Course Outline

Complex Analysis

Math Spring Berkeley CA

C McMullen

Texts

Nehari Conformal Mapping

Ahlfors Lectures on Quasiconformal Mappings

Also recommended

Ahlfors Complex Analysis

Conway Functions of One Complex Variable

Lehto Univalent Functions and Teichmuller Theory

Background in real analysis and basic dierential topology such as

covering spaces and dierential forms is a prerequisite

Relations of complex analysis to other elds include algebraic geome

try complex manifolds several complex variables Lie groups and ho

mogeneous spaces C H

b

C geometry Platonic solids hyperbolic ge

ometry in dimensions two and three Teichmuller theory elliptic curves

and algebraic number theory s and prime numbers dynamics iter

ated rational maps

Algebraic origins of complex analysis solving cubic equations x

ax

b by Tchirnhaus transformation to make a This is done by

introducing a new variable y cx

d such that

P

y

i

P

y

i

even

when a and b are real it may be necessary to choose c complex the

discriminant of the equation for c is b

a

It is negative when

the cubic has only one real root this can be checked by looking at the

product of the values of the cubic at its max and min

Elements of complex analysis C Ri z and related examples for

Q

p

Geometry of multiplication why is it conformal Because

jabj jajjbj so triangles are mapped to similar triangles Visualizing

e

z

lim zn

n

A pie slice centered at n and with angle n is

mapped to the upper semicircle in the limit we nd expi

Denition fz is analytic if f

z exists Note we do not require

continuity of f

Cauchys theorem

R

fzdz Plausibility

R

z

n

dz

R

b

a

t

n

tdt

R

b

a

t

n

n

dt

Proof fzdz is a closed form when f is holomorphic assuming

fz is smooth Discussion f is holomorphic i idfdx dfdy from

this dfdz Moreover dfdx dfdz where ddz ddx

iddy We have f analytic if and only if dfdz Then df

dfdzdz dfdzdz and we see dfdz i dfdz i f is

holomorphic

Proof Goursat assuming only complex dierentiability

Analyticity and power series The fundamental integral

R

dzz The

fundamental power series z

P

z

n

Put these together with

Cauchys theorem

fz

i

Z

fd

z

to get a power series

Theorem fz

P

a

n

z

n

has a singularity where it cannot be analyti

cally continued on its circle of convergence jzj R lim sup ja

n

j

n

Innite products

Q

a

n

converges if

P

ja

n

j The proof is in

two steps rst show that when

Q

ja

n

j converges the dierences in

successive partial products bound the dierences for

Q

a

n

Then

show

P

ja

n

j

Q

ja

n

j exp

P

ja

n

j

Example evaluation of

Q

p

where p ranges over

the primes converges to

Cauchys bound jf

n

j nMRR

n

Liouvilles theorem algebraic

completeness of C Compactness of bounded functions in the uniform

topology Parsevals inequality which implies Cauchys

X

ja

n

j

R

Z

jZjR

jfzj

d MR

Moreras theorem converse to Cauchys theorem Denition of logz

R

z

d Analytic continuation natural boundaries

P

a

n

z

n

Laurent

series fz

P

a

n

z

n

where a

n

i

R

C

fzz

n

dz Classi

cation of isolated singularities removability of singularities of bounded

functions Behavior near an essential singularity WeierstrassCasorati

fU C

Generating functions and

P

F

n

z

n

F

n

the nth Fibonacci number A

power series represents a rational function i its coecients satisy a

recurrence relation Pisot numbers the golden ratio and why are

and such pleasant times

Kroneckers theorem one need only check that the determinants of the

matrices a

iij

i j n are zero for all n suciently large

Residue theorem and evaluation of denite integrals Three types

R

Rcos sin d

R

Rxdx and

R

x

a

Rxdx a R a

rational function

Hardys paper on

R

sinxx dx

The argument principle number of zeros number of poles is equal to

i

Z

f

f

d

Similary the weighted sum of gz over the zeros and poles is given by

multiplying the integrand by g

Winding numbers of the topological nature of the argument principle

if a continuous f C has nonzero winding number on the circle

then f has a zero in the disk

Rouches theorem if jgj jf j on then f g and g have the same

number of zerospoles in Example z

z has all zeros of

modulus less than but only one of modulus less than

Open mapping theorem if f is nonconstant then it sends open sets to

open sets Cor the maximum principle jf j achieves its maximum on

the boundary

Invertibility a If f U V is injective and analytic then f

is analytic b If f

z then f is locally injective at z Formal

inversion of power series

Phragmen Lindelof type results if f is bounded in a strip fa

Imz bg and f is continuous on the boundary then the sup on

the boundary is the sup on the interior

Hadamards circles theorem if f is analytic in an annulus then

logMr is a convex function of log r where Mr is the sup of jf j over

jzj r Proof a function s of one real variable is convex if and only

if s ar satises the maximum principle for any constant a This

holds for logMexps

by considering fzz

a

locally

The concept of a Riemann surface the notion of isomorphism the three

simplyconnected Riemann surface C

b

C and H

A nonconstant map between compact surfaces is surjective by the open

mapping theorem

Theorem AutC faz bg

The complex plane C The notion of metric zjdzj The automor

phism group is solvable Inducing a metric jdzjjzj on C

The cone

metric jdzjj

p

zj giving the quotient by z

The Riemann sphere

b

C and its automorphisms Theorem Aut

b

C

PSL

C A particularly nice realization of this action is as the pro

jectivization of the linear action on C

Mobius transformations invertible form a group act by automor

phisms of

b

C triplytransitive sends circles to circles Proof of last a

circle x

y

Ax By C is also given by r

rA cos

B sin C and it is easy to transform the latter under z z

which replaces r by and by

Classication of Mobius transformations and their trace squared a

identity b parabolic a single xed point c elliptic two xed

points derivative of modulus one

d hyperbolic two xed

points one attracting and one repelling C

Stereographic projection preserves circles and angles Proof for angles

given an angle on the sphere construct a pair of circles through the

north pole meeting at that angle These circles meet in the same angle

at the pole on the other hand each circle is the intersection of the

sphere with a plane These planes meet C in the same angle they meet

a plane tangent to the sphere at the north pole QED

Four views of

b

C the extended complex plane the Riemann sphere the

Riemann surface obtained by gluing together two disks with z z

the projective plane for C

The spherical metric jdzj jzj

Derive from the fact Riemann

circle! and the map x tan and conformality of stereographic

projection

Some topology of projective spaces RP

is the union of a disk and a

Mobius band the Hopf map S

S

GaussBonnet for spherical triangles area equals angle defect Prove

by looking at the three lunes of area for the three angles of a

triangle General form X

R

X

K

R

X

k

Theorem AutH PSL

R Schwarz Lemma and automorphisms

of the disk The hyperbolic metric jdzj Imz on H and its equivalence

to jdzj jzj

on

Classication of automorphisms of H

according to translation dis

tance

Hyperbolic geometry geodesics are circles perpendicular to the circle

at innity Euclids fth postulate given a line and a point not on the

line there is a unique parallel through the point Here two lines are

parallel if they are disjoint

GaussBonnet in hyperbolic geometry a Area of an ideal triangle is

R

R

p

x

y

dydx b Area A of a triangle with two ideal

vertices and one external angle is additive A AA

as a diagram shows Thus A c Finally one can extend

the edges of a general triangle T in a spiral fashion to obtain an ideal

triangle containing T and other triangles each with ideal vertices

Harmonic functions A realvalue function uz is harmonic i u is

locally the real part of an analytic function indeed harmonic means

ddu and v

R

du Harmonic functions are preserved under

analytic mappings Examples electric potential "uid "ow around a

cylinder

The meanvalue principle u average over the circle follows from

Cauchys formula as does the Poisson integral formula up visual

average of u

The Schwarz re"ection principle if U U

and f is analytic on UH

continuous and real on the boundary then fz extends f to all of U

This is easy from Moreras theorem A better version only requires

that Imf at the real axis and can be formulated in terms of

harmonic functions cf Ahlfors

If v is harmonic on U H and vanishes on the real axis then vz

vz extends v to a harmonic function on U For the proof use the

Poisson integral to replace v with a harmonic function on any disk

centered on the real axis the result coincides with v on the boundary

of the disk and on the diameter where it vanishes by symmetry so

by the maximum principle it is v

Re"ection gives another proof that all automorphisms of the disk ex

tend to the sphere

Normal families any bounded family of analytic functions is normal

by ArzelaAscoli

Riemann mapping theorem given a simplyconnected region U C

U C and a basepoint u U there is a unique conformal homeo

morphism f U u such that f

u Proof let F be the

family of univalent maps U u Using a squareroot and an

inversion show F is nonempty Also F is closed under limits By the

Schwarz Lemma jf

uj has a nite maximum over all f F Let f

be a maximizing function If f is not surjective to the disk then we

can apply a suitable composition of a squareroot and two automor

phisms of the disk to get a g F with jg

uj jf

uj again using

the Schwarz Lemma QED

Uniformization of annuli any doublyconnected region in the sphere is

conformal isomorphic to C

or AR fz jzj Rg The

map from H to AR is z z

where logRi The deck

transformation is given by z z where

logR

The class S of univalent maps f C such that f and

f

Compactness of S The Bieberbach Conjecturede Brange

Theorem fz

P

a

n

z

n

with ja

n

j n

The area theorem if fz z

P

b

n

z

n

is univalent on fz jzj

g then

P

njb

n

j

The proof is by integrating fdf over the unit

circle and observing that the result is proportional to the area of the

complement of the image of f

Proof that ja

j rst apply the area theorem to conclude ja

a

j

Then consider

q

fz

for f S

The Koebe Theorem if f S then f Proof if

w is omitted from the image then fz fzw S now apply

ja

j

Riemann surfaces and holomorphic forms The naturality of the

residue and of df

The Residue Theorem the sum of the residues of a meromorphic

form on a compact Riemann surface is zero Application to dff and

thereby to the degree of a meromorphic function

Remarks on forms a holomorphic form on the sphere is zero be

cause it integrates to a global analytic function Moreover a meromor

phic form on the sphere always has more poles than zeros

The SchwarzChristoel formula Let f H U be the Riemann

mapping to a polygon with vertices p

i

i n and exterior angles

i

Then

fz

Z

d

Q

n

q

i

i

d

where fq

i

p

i

Proof compute the nonlinearity Nf f

zf

zdz By Schwarz

re"ection it extends to a meromorphic form on the sphere with simple

poles at the q

i

Using the fact that z

i

straightens out the ith vertex

of U one nds that Nf has residue

i

at q

i

Since a form is

determined by its singularities we have

Nf

X

i

dz

z q

i

and the formula results by integration

Examples of SchwarzChristoel logz

R

d maps to a bigon

with external angles of

sin

z

R

d

p

maps to a triangle

with external angles and

The lengtharea method Let f Ra b Q be a conformal map of a

rectangle to a Jordan region Q C where Ra b a b C

Then there is a horizontal line a fyg whose image has length

L

ab areaQ Similarly for vertical lines

Corollary given any quadrilateral Q the product of the minimum

distances between opposite sides is a lower bound for areaQ

Theorem The Riemann map to a Jordan domain extends to a home

omorphism on the closed disk Proof given a point z map

to an innite strip sending z to one end Then there is a sequence of

disjoint squares in the strip tending towards that end The images of

these squares have areas tending to zero so there are crosscuts whose

lengths tend to zero as well by the lengtharea inequality This gives

continuity at z Injectivity is by contradiction if the map is not in

jective then it is constant on some interval along the boundary of the

disk

WeierstrassHadamard factorization theory A good reference for this

material is Titchmarsh Theory of Functions We will examine the

extent to which an entire function is determined by its zeros We

beginning by showing that any discrete set in C arises as the zeros of

an entire function

Weierstrass factor Inspired by the fact that z exp log z

and that log z z z

z

we set

E

p

z z exp

z

z

z

p

p

By convention E

z z

Theorem For jzj we have jE

p

z j jzj

p

Proof Writing E

p

z

P

a

k

z

k

one may check by computing

E

p

z

that all a

k

P

ja

k

j and a

a

a

p

Then

jE

p

z j j

P

a

k

z

k

j jzj

p

P

ja

k

j jzj

p

QED

Theorem If

P

rja

n

j

p

n

for all r then fz

Q

E

p

n

za

n

converges to an entire function with zeros exactly at the a

n

Cor Since p

n

n works for any a

n

we have shown any discrete

set arises as the zeros of an entire function

Blaschke products Let f be a proper map of degree d Then

fz e

i

d

Y

z a

i

a

i

z

where the a

i

enumerate the zeros of f

Jensens formula Let fz be holomorphic on the disk of radius R

about the origin Then the average of log jfzj over the circle of radius

R is given by

log jfj

X

fz jzjR

log

R

jzj

Proof Suces to assume R Clear if f has no zeros because

log jfzj is harmonic Clear for a Blaschke factor zaaz But

the formula is true for fg if it is true for f and g so we are done

Cor Let nr be the number of zeros of f inside the circle of radius r

Then

Z

R

nr

dr

r

Z

log jfRe

i

jd log jfj

Remark We have used the mean value property of harmonic functions

This holds for any harmonic function u on the disk by writing u

Ref f holomorphic and then applying Cauchys integral formula for

f

The physical idea of Jensens formula is that log jf j is the potential for

a set of unit point charges at the zeros of f

Entire functions of nite order An entire function f C C is of

nite order if there is an A such that jfzj Oexp jzj

A

The

least such A is the order of f

Examples Polynomials have order sinz cosz expz have order

cos

p

z has order expexpz

has innite order

Number of zeros By Jensens formula if f has order then nr

Or

where nr is the zero counting function for f Corollary

P

ja

i

j

where a

i

enumerates the zeros of f other than zero

itself

In other words a

i

f where a

i

is the exponent of convergence

of the zeros of f ie the least such that

P

ja

i

j

Denition a canonical product is an entire function of the form

fz z

m

Y

E

p

za

i

where p is the least integer such that

P

jza

i

j

p

for all z

Hadamards Factorization Theorem Let f be an entire function of

order Then fz P z expQz where P is a canonical product

with the same zeros as f and Q is a polynomial of degree less than or

equal to

To prove Hadamards theorem we develop two estimates First we

show a canonical product P z is an entire function of order a

i

This is the least order possible for the given zeros by Jensens theorem

Second we show a canonical product has mr expjzj

for most

radii r where mr is the minimum of jfzj on the circle of radius r

More precisely for any this lower bound holds for all r outside

a set of nite total length In particular it holds for r arbitrarily large

Given these facts we observe that fzP z is an entire function of

order with no zeros Thus Qz log fz is an entire function

satisfying ReQz O jzj

This implies Q is a polynomial

Indeed for any entire function fz

P

a

n

z

n

we have ja

n

jr

n

maxAr Ref

where Ar maxRe fz over the cir

cle of radius r

Example

sinz z

Y

n

z

n

Indeed the right hand side is a canonical product and sinz has

order one so the formula is correct up to a factor expQz where Qz

has degree one But since sinz is odd we conclude Q has degree

zero and by checking the derivative at z of both sides we get

Q

Little Picard Theorem A nonconstant entire function omits at most

one value in the complex plane This is sharp as shown by the example

of expz

Great Picard Theorem Near an essential singularity a meromorphic

function assumes all values on

b

C with at most two exceptions

Proof of Little Picard The key fact is that the universal cover of

C f g can be identied with the upper halfplane This can be seen

by constructing a Riemann mapping from the upper halfplane to an

ideal hyperbolic triangle sending and to the vertices and then

developing both the domain and range by Schwarz re"ection Indeed

with suitable normalizations we have that C f g is isomorphic to

H # where # PSL

Z is the group of matrices congruent to

the identity modulo This is a free group on two generators

Given this fact we lift an entire function f C C f g to a map

e

f C H which is constant by Liouvilles theorem because H

Proof of Great Picard Let f

C be a holomorphic

function omitting three values on the sphere normalized to be zero

one and innity Consider a loop around the puncture of the disk If

f sends to a contractible loop on the triplypunctured sphere then f

lifts to a map into the universal cover H which implies by Riemanns

removability theorem that f extends holomorphically over the origin

Otherwise by the Schwarz lemma f is a homotopy class that can

be represented by an arbitrarily short loop Thus it corresponds to a

puncture which we can normalize to be z rather than or

It follows that f is bounded near z so again the singularity is not

essential

The Gamma Function

#z

expz

z

Y

z

n

expzn

Note that this expression contains the reciprocal of the canonical prod

uct associated to the nonpositive integers The constant is chosen

so that # it can be given by

lim

n

X

k logn

This expression is the error in an approximation to

R

n

dxx by the

area of n rectangles of base one lying over the graph

Gausss formula

#z lim

n

nn

z

zz z n

The functional equation

#z z#z

Corollary #n n

The integral representation for Rez

#z

Z

e

t

t

z

dt

t

In other words #z is the Mellin transform of the function e

t

on R

The Mellin transform is an integral against characters R

C

given by t t

z

and as such it can be compared to the Fourier

transform for the group R under addition and to Gauss sums Indeed

the Gauss sum

X

np

ne

inp

is the analogue of the Gamma function for the group Zp

Relation to sine function

sinz

#z# z

Proof Form the product #z#z and use the functional equation

Periodic functions If f C X has period then fz F expiz

where F C

X Example

X

z n

sin

z

From this we get

P

n

Elliptic doublyperiodic functions Such functions can be considered

as holomorphic maps f X

b

C where X C $ By general easy

results on compact Riemann surfaces An entire elliptic function is

constant The sum of the residues of f over X is zero The map f has

as many zeros as poles

More interesting is the fact that if f is nonconstant with zeros a

i

and

poles p

i

then

P

a

i

P

p

i

in the group law on X Proof integrate

zdff around a fundamental parallelogram in C

We will see that we may construct an elliptic function with given zeros

and poles subject only to this constraint

Construction of elliptic functions For n

n

z

X

z

n

denes an elliptic function of degree n For degree two we adjust the

factors in the sum so they vanish at the origin and obtain the denition

of the Weierstrass function

z

z

X

z

To see z is elliptic use the fact that z z and

z

z which implies z z A

Other ways to construct elliptic functions of degree two if $ is gener

ated by

rst sum over one period to get

f

z

X

z n

sinz

then we have

z

X

f

z n

and the convergence is rapid Similarly if we write X C

z

z then

fz

X

n

z

n

z

converges to an elliptic function of order two

Laurent expansion and dierential equation Expanding z

in

a Laurent series about z and summing we obtain

z

z

X

n z

n

G

n

where

G

n

X

n

By a straightforward calculation using the fact that an elliptic function

with no pole is constant we have

z

z

g

z g

where g

G

and g

G

Geometry of the Weierstrass map X

b

C The critical points of

are the points of order two the critical values are innity and the zeros

of the cubic equation px x

g

x g

These are distinct

The map z z

z

sends X fg to the ane cubic curve

y

px Thus uniformizes! this plane curve of genus

When $ has a rectangular fundamental domain the function can

also be constructed by taking a Riemann mapping from this rectangle

with dimensions reduced by a factor of two to the upper halfplane

and developing by Schwarz re"ection Then the dierential equation

for comes from the SchwarzChristoel formula

Function elds Given any Riemann surface X the meromorphic func

tions on X form a eld KX Then K is a contravariant functor from

category of Riemann surfaces with nonconstant maps to the category

of elds with extensions Example K

b

C C z

Theorem For X C $ KX C x yy

x

g

x g

To see that and

generate KX is easy Any even function f

X

b

C factors through fz F z

and so lies in C Any

odd function becomes even when multiplied by

and any function is

a sum of one even and one odd

To see that the eld is exactly that given is also easy It amounts

to showing that KX is of degree exactly two over C and is

transcendental over C The rst assertion is obvious and if the second

fails we would have KX C which is impossible because is

even and

is odd

An elliptic function with given poles and zeros It is natural to try to

construct an elliptic function by forming the Weierstrass product for

the lattice $

z z

Y

z

exp

z

z

Then

zz

z from which it follows that z

z expa

b

z From this it is easy to see that

z a

z a

n

z p

z p

n

denes an elliptic function whenever

P

a

i

P

p

i

and therefore this

is the only condition imposed on the zeros and poles of an elliptic

function

The moduli space of elliptic curves M

classies Riemann surfaces

of genus up to isomorphism Since any such surface is given by

X C $ and we may normalize so $ Z Z where H it is

not hard to see

M

H PSL

Z

Similarly an elliptic curve with a labelling of its points of order two is

classied by M

H #

We now present another proof that H # is naturally isomorphic to

b

C f g Given $ we get the cubic polynomial x

g

x g

xx

xx

xx

whose roots are exactly the values of z at

the points of order two Thus the crossratio x

x

x

x

depends only on the location of in M

and thus we have a map

M

b

C f g

But we can invert this map given form the degree two cover X

of

b

C

branched over f g Once we know X

is isomorphic to C $ for

some $ it is easy to show we have reconstructed up to an element of

# One might appeal to the uniformization theorem here but in fact

it is elementary to see that dx

q

xx x lifts to a nowherezero

holomorphic form on X

This one form gives a complete Euclidean

metric and so the universal cover of X

can be identied with C

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