Complex Analysis: Convergence of Functions and Möbius Transform, Study notes of Complex analysis

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Complex Analysis

Xue-Mei Li

May 16, 2016

Contents

Prologue

This is the lecture notes for the third year undergraduate module: MA3B8. If you need not be motivated, skip this section. Complex Analysis is concerned with the study of complex number valued functions with complex number as domain. Let f : C → C be such a function. What can we say about it? Where do we use such an analysis? The complex number i =

− 1 appears in Fourier Transform, an important tool in analysis and engineering, and in the Schr¨odinger equation,

iℏ

∂ψ ∂t

ℏ^2

2 m

∂^2 ψ ∂^2 x

• V (x)ψ(t, x),

a fundamental equation of physics, that describes how a wave function of a physical system evolves. Complex Differentiation is a very important concept, this is allured to by the fact that a number of terminologies are associated with ‘complex differentiable’. A function, complex differentiable on its domain, has two other names: a holomor- phic map and an analytic function, reflecting the original approach. The first meant the function is complex differentiable at every point, and the latter refers to functions with a power series expansion at every point. The beauty is that the two concepts are equivalent. A complex valued function defined on the whole complex domain is an entire function. Quotients of entire functions are Meromorphic functions on the whole plane. A map is conformal at a point if it preserves the angle between two tangent vec- tors at that point. A complex differentiable function is conformal at any point where its derivative does not vanish. Bi-holomorphic functions, a bi-jective holomorphic function between two regions, are conformal in the sense they preserve angles. Of- ten by conformal maps people mean bi-holomorphic maps. Conformal maps are the building blocks in Conformal Field Theory. It is conjectured that 2 D statistical mod- els at criticality are conformal invariant. An exciting development is SLE, evolved from the Loewner differential equation describing evolutions of conformal maps. The Schramann-Loewner Evolution (also known as Stochastic Loewner Evolution, abbre- viated as SLE ) has been identified to describe the limits of a number of lattice models in statistical mechanics. Two mathematicians, W. Werner and S. Smirnov, have been awarded the Fields medals for their works on and related to SLE. Complex valued functions are built into the definition for Fourier transforms. For f : R → R,

f^ ˆ (k) = √^1 2 π

∞

e−ikxdx, k ∈ R.

Fourier transform extends the concept of Fourier series for period functions, is an im- portant tool in analysis and in image and sound processing, and is widely used in elec- trical engineering.

A well known function in number theory is the Riemann zeta-function,

ζ(s) =

∑^ ∞

n=

ns^

The interests in the Riemann-zeta function began with Euler who discovered that the Riemann zeta function can be related to the study of prime numbers.

ζ(s) = Π

1 − p−s^

The product on the right hand side is over all prime numbers:

1 − p−s^

1 − 2 −s^

1 − 3 −s^

1 − 5 −s^

1 − 7 −s^

1 − 11 −s^

1 − p−s^

The Riemann-zeta function is clearly well defined for s > 1 and extends to all complex numbers except s = 1, a procedure known as the analytic /meromorphic continuation of a real analytic function. Riemann was interested in the following question: how many prime number are below a given number x? Denote this number π(x). Riemann found an explicit formula for π(x) in his 1859 paper in terms of a sum over the zeros of ζ. The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function lie on the critical line s = 12. The Clay institute in Canada has offered a prize of 1 million dollars for solving this problem. In symplectic geometry, symplectic manifolds are often studied together with a complex structure. The space C is a role model for symplectic manifold. A 2-dimensional symplectic manifold is a space that looks locally like a piece of R^2 and has a symplectic form, which we do not define here. We may impose in addition a complex structure Jx at each point of x ∈ M. The complex structure Jx is essentially a matrix s.t. −J x^2 is the identity and defines a complex structure and leads to the concept of Kh¨aler manifolds. Finally we should mention that complex analysis is an important tool in combina- torial enumeration problems: analysis of analytic or meromorphic generating functions provides means for estimating the coefficients of its series expansions and estimates for the size of discrete structures.

Topics

Holomorphic Functions, meromorphic functions, poles, zeros, winding numbers (rota- tion number/index) of a closed curve, closed curves homologous to zero, closed curves homotopic to zero, classification of isolated singularities, analytical continuation, Con- formal mappings, Riemann spheres, special functions and maps. Main Theorems: Goursat’s theorem, Cauchy’s theorem, Cauchy’s derivative formulas, Cauchy’s inte- gral formula for curves homologous to zero, Weirerstrass Theorem, The Argument principle, Rouch´e’s theorem, Open Mapping Theorem, Maximum modulus principle, Schwartz’s lemma, Mantel’s Theorem, H¨urwitz’s theorem, and the Riemann Mapping Theorem.

References

• L. V. Ahlfors, Complex Analysis, Third Edition, Mc Graw-Hill, Inc. (1979)

Chapter 1

Complex Differentiation

1.1 The Complex Plane

The complex plane C = {x + iy : x, y ∈ R} is a field with addition and multiplication, on which is also defined the complex conjugation x + iy = x − iy and modulus (also called absolute value) |z| =

z ¯z =

x^2 + y^2. It is a vector space over R and over C with the norm |z 1 − z 2 |. We will frequently treat C as a metric space, with distance d(z 1 , z 2 ) = |z 1 − z 2 |, and so we understand that a sequence of complex numbers zn converges to a complex number z is meant by that the distance |zn − z| converges to zero. The space C with the above mentioned distance is a complete metric space and so a sequence converges if and only if it is a Cauchy sequence. Since

|zn − z|^2 = |Re(zn) − Re(z)|^2 + |Im(zn) − Im(z)|^2 ,

zn converges to z if and only if the real parts of (zn) converge to the real part of z and the imaginary parts of (zn) converge to the imaginary part of z. In polar Coordinates z ∈ C can be written as z = reiθ^ where r = |z| and θ is a real number, called the argument. We note specially Euler’s formula:

eiθ^ = cos(θ) + i sin(θ),

so arg z is a multi valued function. It is standard to take the principal value −π < Argz ≤ π, a rather arbitrary choice. Since e^2 πik^ = 1 for k an integer, the nth root function is multi-valued. If

ωk = e

2 πkn i , k = 0, 1 ,... , n − 1 ,

the nth roots of the unity, then

(reiθ^ ) n^1 = r

(^1) n ei^

θn ωk.

1.2 Complex Functions of One Variables

To discuss complex differentiation of a function, we request that it is defined on a subset of the complex plane C which is open. By a set we would usually mean a subset of the

complex plane C. A set U is open if about every point in U there is a disc contained entirely in U. We further assume that the set is connected, otherwise we could treat it as a separate function on each connected subset. A subset of C is connected if any two points from the subset can be connected by a continuous curve which lies entirely within the subset.

An open set is connected if and only if it is not disconnected in the sense that it is not the union of two disjoint open sets.

Definition 1.2.1 By a region we mean a connected open subset of C. By a proper region we mean an open connected subset of C that is not the whole complex plane.

From now on, by a function we mean a function f : U → C where U is a region. By an open disc we mean {z : |z − z 0 | < r} where z 0 ∈ C and r > 0. A closed disc is {z : |z −z 0 | ≤ r} where z 0 ∈ C and r ≥ 0. The unit disc centred at 0 is denoted by D = {z : |z| < 1 }.

Other frequently seen open sets are the deleted discs and the annulus

{z : 0 < |z − z 0 | < r}, {z : r 1 < |z − z 0 | < r 2 },

and polygons.

Example 1.2.1 Given z 0 ∈ C, the function f : C → C given by the formula f (z) = z + z 0 is said to be a translation.

As a set we may wish to identify a complex number s + it with the pair of real numbers (s, t), so C is identified with R^2. Since, for z = x + iy and c = s + it,

c(x + iy) = (sx − ty) + i(tx + sy),

the map z → cz is represented by a a linear map: ( x y

s −t t s

x y

Multiplication by i is the same as multiply by J on the left, where

J =

Example 1.2.2 Given c ∈ C, the function f (z) = cz is of the form below. For z = x + iy, c = |c| eiθ^ , ( x y

7 → |c|

cos(θ) − sin(θ) sin(θ) cos(θ)

x y

This is the composition of a rotation by an angle θ and a scaling by |c|. This map preserves the angle between two vectors, i.e. it is a conformal map.

− 1 0

− 1

1

− 1 1

− 1

1

z =

√ w

w = z

Figure 1.2: Graph by E. Hairer and G. Wanner

Example 1.2.4 Define f on C \ (−∞, 0] by f (w) =

w, the principal brach of the square root function. So f (reiθ^ ) =

reiθ/^2 , −π < θ < π. It has another formula:

f (w) =

|w|ei(Argw/2), w ∈ C \ (−∞, 0].

It maps the slit w plane into the right half of the z-plane. The other branch of the square root is −

w. It is possible to glue the two slit domains together to form a complex manifold, known as a Riemann surface, so in one sheet (chart) the function takes the value of one brach and in the other we use the other brach in a way f changes continuously as w changes.

Example 1.2.5 The map f (z) = (^1) z , the inversion map, is defined on C \ { 0 }. It is easy to see that f takes circles centred at the origin to circles centres at the origin. It take the locus of the solutions of |z − z 0 | = r to that of a circle in the w-plane, see Proposition 2.2.6 and Example sheets.

Example 1.2.6 (M¨obius Transforms) Let a, b, c, d ∈ C where ad 6 = bc. Define

f (z) =

az + b cz + d

If c = 0 the domain of f is C otherwise it is C \ {− dc }.

Exercise 1.2.7 1. Prove that for any real number r, not 1 , the equation

|z − z 1 | = r|z − z 2 |

determines a circle.

2. Prove that any M¨obius transform is a composition of translations, scalings, and inversions.

Later we will see that M¨obius Transforms can be considered as maps on the extended complex plane, the Riemann sphere.

1.3 Complex Linear Functions

We identify R^2 with C. A function T : R^2 → R^2 is real linear if for all z 1 , z 2 , z ∈ R^2 ,

T (z 1 + z 2 ) = T (z 1 ) + T (z 2 ), T (rz) = rT (z), ∀r ∈ R

A map T : C → C is complex linear if for all z 1 , z 2 , z ∈ C,

T (z 1 + z 2 ) = T (z 1 ) + T (z 2 ), T (kz) = kT (z), ∀k ∈ C

Proposition 1.3.1 A real linear function T : R^2 → R^2 is complex linear iff

T (i) = iT (1).

We now look at the matrix representations. Every real linear map is of the form ( x y

a b c d

x y

If k = s + it, the complex linear map T (z) = kz is given by

T

x y

s −t t s

x y

For every real linear map T there exists a unique pair of complex numbers λ and μ such that T (z) = λz + μz,¯

which is complex linear if and only if μ = 0. Furthermore,

λ =

(a + ic) +

i (b + id)

, μ =

(a + ic) −

i (b + id)

1.4 Complex Differentiation

Let f = u + iv, defined in a region U. When C is identified as R^2 we may treat u and v as real valued functions on R^2. In this way f is an R^2 valued function of two real variables x and y. Then f is (real) differentiable at (x 0 , y 0 ) if there exists a linear map (df )(x 0 ,y 0 ) : R^2 → R^2 and a function φ such that

f (x, y) = f (x 0 , y 0 ) + (df )(x 0 ,y 0 )

x − x 0 y − y 0

• φ(x, y)

x − x 0 y − y 0

Figure 1.3: Handwriting by Riemann

Theorem 1.4.3 1. If f : U → R is complex differentiable at z 0 = x 0 + iy 0 then f is real differentiable at (x 0 , y 0 ) and the Cauchy-Riemann Equations hold at z 0 :

∂xu = ∂y v, ∂y u = −∂xv. (1.4.4)

Also, f ′(z 0 ) = ∂xu + i∂xv =

i

(∂y u + i∂y v).

2. If f : U → C is real differentiable and satisfies the Cauchy-Riemann equation at a point (x 0 , y 0 ) ∈ U then f is complex differentiable at z 0 = x 0 + iy 0. In particular, if u, v : R^2 → R^2 are C^1 functions satisfying the Cauchy-Riemann equations in U then f = u + iv is complex differentiable in U.

Proof (1) Write f ′(z 0 ) = s + it. Then by the definition, (1.4.3),

f (z) = f (z 0 ) +

s −t t s

(z − z 0 ) + ψ(z) |z − z 0 |.

This is (1.4.1) with

(df )(x 0 ,y 0 ) =

s −t t s

So f is real differentiable with ( ∂xu ∂y u ∂xv ∂y v

s −t t s

Thus the Cauchy-Riemann equation follows and

f ′(z 0 ) = s + it = ∂xu + i∂xv = ∂y v − i∂y u.

(2) We have (1.4.1),

f (x, y) = f (x 0 , y 0 ) + (df )(x 0 ,y 0 )

x − x 0 y − y 0

• φ(x, y)

x − x 0 y − y 0

By the Cauchy-Riemann equation the Jacobian matrix is the following form

J =

∂xu −∂xv ∂xv ∂xu

and represent the complex linear map: multiplication by f ′(z 0 ) := ∂xu + i∂xv, Hence

f (z) = f (z 0 ) + f ′(z 0 )(z − z 0 ) + φ(z) |z − z 0 |.

This implies f is complex differentiable at z 0. If u, v are C^1 , then f = (u, v) is differentiable and the previous statement applies. 

The Cauchy Riemann equation can also be written as ∂xf = (^1) i ∂y f.

1.5 The ∂ and ∂¯ operator

Given a function f , we have

f (x, y) = f

z + ¯z 2

z − z¯ 2 i

This inspires the notation :

∂z =

(∂x +

i

∂y ), ∂¯z =

(∂x −

i

∂y ). (1.5.1)

It is common to denote ∂z by ∂ and ∂¯z by ∂¯. It is clear that ∂z¯ f = 0 is the Cauchy- Riemann equation We can reformulate the earlier theorem using these notations. Suppose that f is complex differentiable at z then f is real differentiable at z and,

∂f^ ¯ (z) = 0, f ′(z) = ∂z f (z).

1.6 Harmonic Functions

Definition 1.6.1 A real valued function u : R^2 → R is a harmonic function if ∆u = 0 where ∆ = ∂xx + ∂yy is the Laplacian.

Proposition 1.6.1 If u, v are C^2 functions and satisfies the Cauchy-Riemann equations

∂xu = ∂y v, ∂y u = −∂xv,

then u, v are harmonic functions. Consequently u, v are C∞.

Proof We differentiate the Cauchy-Riemann equation to see

∂xxu = ∂xy v, ∂yy u = −∂yxv.

Consequently ∂xxu + ∂yy u = 0. Similarly, ∂xxv + ∂yy v = 0. From standard theory in PDE, a solution of the elliptic equation ∆u = 0 is C∞. 

Later we see that if f is differentiable in a region, it has derivatives of all orders. So the conditions u, v ∈ C^2 can be reduced to C^1.

then so is its inverse:

J−^1 =

det J

∂xu −∂y u ∂y u ∂xu

Since f ′(z 0 ) = ∂xu(z 0 ) − i∂y u(z 0 ), and

1 f ′(z 0 )

∂xu(z 0 ) + i∂y u(z 0 ) (∂xu)^2 (z 0 ) + (∂y u(z 0 ))^2

det J(z 0 )

(∂xu(z 0 ) + i∂y u(z 0 )).

In conclusion, J(z 0 ) represents f ′(z 0 ), J−^1 (z 0 ) represents (^) f ′(^1 z 0 ). This leads to the following theorem.

Theorem 1.8.3 Suppose that f : U → C is complex differentiable and u, v have continuous partial derivatives. Suppose f ′(z 0 ) 6 = 0 for some z 0 ∈ U.

• Then there exists a disc U around z 0 such that f : U → f (U ) is a bijection, f (U ) is open and f −^1 : f (U ) → U is continuous.
• Furthermore f −^1 is complex differentiable on f (U ) and

(f −^1 )′(f (z)) =

f ′(z)

, z ∈ U.

Proof The first part of the statement follows from real analysis. To see f −^1 is differ- entiable, write w 0 = f (z 0 ), w = f (z). Since f and f −^1 are continuous, w → w 0 is equivalent to z → z 0. Since f ′(z 0 ) 6 = 0,

f −^1 (w) − f −^1 (w 0 ) w − w 0

f (z)−f (z 0 ) z−z 0

Take w → w 0 we see that the limit on the left hand side exists and

f −^1 (w 0 ) =

f ′(z 0 )

f ′(f −^1 (w))

Remark 1.8.4 Later we see that if f is complex differentiable, it is infinitely differen- tiable. If f is one to one then f ′^ does not vanish. See section 8.3.

Chapter 2

The Riemann Sphere and

M¨obius Transforms

2.1 Conformal Mappings

Definition 2.1.1 A parameterized curve in the complex plane is a function z : [a, b] → C where [a, b] is closed interval of R.

If z(t) = x(t) + iy(t) its derivative is z′(t) = x′(t) + iy′(t).

Definition 2.1.2 The parameterized curve : [a, b] → C is smooth if z′(t) exists and is continuous on [a, b]. We assume furthermore z′(t) 6 = 0.

The derivatives at the ends are understood to be one sided derivatives. From now on by a curve we mean a smooth curve. If z′(t) does not vanish the curve has a tangent at this point, whose direction is determined by arg(z′(t)).

Definition 2.1.3 Let z 1 : [a 1 , b 1 ] → C and z 2 : [a 2 , b 2 ] → C be two smooth curves intersecting at z 0. The angle of the two curves is the angle of their derivatives at this point.

They are given by the difference of the arguments of their derivatives. If z 1 (t 1 ) = z 2 (t 2 ) = z 0 , their angle at the point z 0 is: arg(z′ 2 (t 2 )) − arg(z′ 1 (t 1 )).

Definition 2.1.4 A map f : U → C is conformal at z 0 if it preserves angles, i.e. if z 1 and z 2 are two curves meeting at z 0 , the angle from f ◦ z 1 to f ◦ z 2 at f (z 0 ) are the same as the angle from z 1 to z 2 at z 0.

Example 2.1.1 The linear map f (z) = kz where k 6 = 0, is a conformal map as it is composed of scaling by |k| and rotating by the angle arg(k). c.f.Example 1.2.

Example 2.1.2 f (z) = ¯z is not a conformal map. This map reverses orientation.

Theorem 2.1.3 If f : U → C is holomorphic at z 0 and f ′(z 0 ) 6 = 0, then f is conformal at z 0.

2.2.1 The Extended Complex Plane

To make the statements neat we add a point at infinity to C and define the extended complex plane to be C∗^ = C ∪ {∞}

with the convention:

1 0

= 0, a + ∞ = ∞, a − ∞ = ∞,

and for a 6 = 0, a · ∞ = ∞ · a = ∞.

2.2.2 Properties of M¨obius Transforms

Let f (z) = azcz++bd. We extend the M¨obius transform f from C to C∗^ by defining:

f (−

d c

) = ∞, f (∞) =

a c

if c 6 = 0.

If c = 0, f (z) = (^1) f (az + b) is defined on the whole plane, then we define

f (∞) = ∞, if c = 0.

The function f has an inverse

f −^1 (w) =

dw − b −cw + a

Note that multiply a, b, c, d, by a non-zero number λ does not change the function

f (z) =

az + b cz + d

azλ + bλ cλz + dλ

Hence we may eliminate one parameter and assume that ad − bc = 1. We define

M =

az + b cz + d

: ad − bc = 1, a, b, c, d, ∈ C

Theorem 2.2.1 The set M of M¨obius transforms is a group under composition. Each M¨obius transform is a composition of the following maps:

(1) translation: z 7 → z + a for some complex number a;

(2) composition of scaling and rotation:

z 7 → kz, some k ∈ C, k 6 = 0.

(3) Inversion: z 7 → (^1) z.

Proof For the group we check the following:

• f (z) = z is the identity. (a = 1, b = c = 0, d = 1)
• If f (z) = azcz++db ∈ M, then

f −^1 (w) =

dw − b −cw + a

∈ M.

• If f (z) = azcz++db ∈ M and g(z) = ¯az+ ¯b ¯cz+ d¯ ∈ M. Then^ f^ ◦^ g^ =^

Az+B Cz+D ∈ M^ where the complex numbers A, B, C, D are given by ( A B C D

a b c d

¯a ¯b ¯c d¯

For the second part of the statement, if c = 0, azd+ b= ad z + (^) db. If c 6 = 0,

f (z) =

az + b cz + d

a c

bc − ad c^2

z + dc

Example 2.2.2 The map f (z) = z z+1− 1 is called the Cayley transform. It takes C \ { 1 } to itself, f : C \ { 1 } → C \ { 1 } is a bijection and f −^1 = f. Let us consider f as a map on C∗^ by setting f (1) = ∞, f (∞) = 1. Note

f (x + iy) =

x^2 + y^2 − 1 (x − 1)^2 + y^2

− 2 y (x − 1)^2 + y^2 i.

Let γ = {x^2 + y^2 = 1} with γ+^ and γ−^ denote respectively the upper and lower half of the circle. Then,

• f sends {− 1 , 0 , 1 } to { 0 , − 1 , ∞} respectively.
• f sends the upper circle to the lower half of the imaginary axis.
• f sends x-axis to x-axis. It send the x-axis within the unit disc to the negative x-axis.
• f sends the upper half of the unit disc to the third quadrant.
• f sends the lower circle to the upper half of the imaginary axis.
• f sends the lower half of the unit disc to the second quadrant.
• f sends the exterior of the unit circle to the right half of the plane.

If z 2 , z 3 , z 4 are distinctive points in C∗^ we associate to it the M¨obius transform

f (z) =

z − z 3 z − z 4

z 2 − z 3 z 2 − z 4

(z − z 3 )(z 2 − z 4 ) (z − z 4 )(z 2 − z 3 )

, if z 2 , z 3 , z 4 ∈ C.

If one of these points is the point at infinity the map is interpreted as following:

f (z) =

z − z 3 z − z 4 , if z 2 = ∞ z 2 − z 4 z − z 4

, if z 3 = ∞

z − z 3 z 2 − z 3

, if z 4 = ∞.

Note that if z 2 , z 3 , z 4 ∈ C,

f (z 2 ) = 1, f (z 3 ) = 0, f (z 4 ) = ∞.