1The complex numbers - math.tu-berlin.de · Functions u and v that are the real and imaginary part...

B. Springborn Complex Analysis I Summary (v 17.7.) Summer 2017

1 The complex numbers 1©

• What are the real numbers?� axiomatic characterization� constructions

• What are the complex numbers?� If F is any field that contains R and an element i with i2 = −1, then the set of numbers C ={a+ bi | a, b ∈ R} is a subfield of F .

� If C= {a+bi} is another field obtained this way, then there is a unique field isomorphism C→ Cmapping i to i and every real number to itself.

� The field of complex numbers is uniquely characterized as the smallest field containing R andan imaginary unit i.

� In fact: Every finite dimensional field extension of R is isomorphic to C. [without proof]� Constructive approach: The complex numbers C are the real vector space R2 equipped with

multiplication(x1, y1) · (x2, y2) = (x1 x2 − y1 y2, x1 y2 + x2 y1).

We write i for (0,1) and x for (x , 0). From now on, we will use this as the definition of C.• Collection of some stuff that should be known:� Real and imaginary parts, Re z and Im z� complex conjugation z 7→ z� Re z = 1

2 (z + z), Im z = 12i (z − z)

� polar representation: z = r(cosφ + i sinφ) where r > 0, φ ∈ R� absolute value (or modulus) |z|=

p

(Re z)2 + (Im z)2 =p

zz� argument φ determined up to multiples of 2π (if z 6= 0)� Euler’s formula: ei y = cos y + i sin y� ex+i y = ex(cos y + i sin y)� z = elog |z|+iφ

� geometric interpretation of complex multiplication� complex multiplication as R-linear map

Literature Many good books cover the material of this course. I mostly follow

• Ahlfors. Complex Analysis• Jänich. Funktionentheorie• Dirk Ferus’ lecture notes, http://page.math.tu-berlin.de/~ferus/KA/komplexeAnalysis.pdf

2 Complex differentiation

Definition 2.1. Let U ∈ C be an open subset, z0 ∈ U . A function f : U → C is called [complex]differentiable in z0, if the limit

limz→z0

f (z)− f (z0)z − z0

=: f ′(z0)

exists. In this case, f ′(z0) is called the [complex] derivative of f at z0. If f is differentiable at all pointsz0 ∈ U , then f is called holomorphic (or [complex] analytic) on U . A holomorphic function on C iscalled an entire function [ganze Funktion].

• same definition of differentiability and derivative as in Analysis I, only R replaced by C

1

http://page.math.tu-berlin.de/~ferus/KA/komplexeAnalysis.pdf


• Basic rules for differentiation remain valid, with same proofs• Examples:� Polynomials p(z) =

∑ni=0 aiz

i are entire functions

� Rational functions f (z) = p(z)q(z) , where p and q are polynomials and q 6= 0, are holomorphic on

U = {z |q(z) 6= 0}.� power series (see following section)

3 Power series 2©

Theorem 3.1. (i) For every power series∑∞

k=0 ak(z − z0)k (with complex coefficients ak) there is anR ∈ [0,∞] (the radius of convergence), such that the series converges absolutely if |z − z0| < R anddiverges if |z − z0|> R.

(ii) In general,R−1 = lim sup

n→∞

nÆ

|an| .

(iii) If the limit on the right hand side exists, then

R−1 = limn→∞

|an+1||an|

.

(iv) If 0< r < R, then the power series converges uniformly on the closed disk�

z�

� |z − z0| ≤ r

.

Proof. Exactly like in real analysis.

Theorem 3.2. If f (z) =∑∞

k=0 ak(z − z0)k has radius of convergence R > 0, then f is holomorphicon the open disk with radius R around z0, and one can differentiate term by term: The power series∑∞

k=1 kak(z − z0)k−1 also has radius of convergence R and represents f ′(z).

Proof. Later, using complex integration.

• Examples� ez

� cos z, sin z� cosh z, sinh z

4 Real and complex differentiability

The function f : U → C is complex differentiable at z0 with derivative f ′(z0) if and only if

f (z0 + h) = f (z0) + f ′(z0)h+ r(h) with limh→0

r(h)h= 0.

The function f is differentiable the real sense at z0 if

f (z0 + h) = f (z0) + d fz0(h) + r(h) with lim

h→0

r(h)‖h‖

= 0.

2


Theorem 4.1. The function f (x , y) =�

u(x , y), v(x , y)�

is complex differentiable in z0 if and only if itis differentiable in the real sense and one (hence both) of the following equivalent conditions hold:

(i) The real derivative

d f =

� dud x

dud y

dvd x

dvd y

�

is C-linear at z0.(ii) The Cauchy–Riemann equations

dud x=

dvd y

,dud y= −

dvd x

are satisfied at z0.

Example 4.2. f (z) = z

Theorem 4.3. If f is holomorphic on an open and connected set U ⊆ C, and if f ′(z) = 0 for all z ∈ U,then f is constant.

Definition 4.4. An open and connected subset of C is called a domain.

Theorem 4.5. If f is holomorphic and real valued on a domain, then f is constant.

Important Example 4.6.

• Principal value argument function, Arg : C \ (−∞, 0]→ (−π,π).• Principal value logarithm function, Log : C \ (−∞, 0]→ C, Log(z) = log |z|+ i Arg z.

Harmonic functions 3©

• Laplace operator ∆= div grad= ∂ 2

∂ x21+ · · ·+ ∂ 2

∂ x2n

• Laplace equation ∆ f = 0• harmonic functions

Theorem 4.7. Real and imaginary parts of a holomorphic [and twice continuously differentiable] func-tion are harmonic.

Remark. We will see later that holomorphic functions are C∞, so the condition in brackets is auto-matically satisfied.

Theorem 4.8. If u : U → R is a harmonic function on a convex/star shaped/simply connected domainU ⊆ C, then there exists a harmonic function v : U → R so that f = u+ iv is holomorphic. The functionv is uniquely determined up to an additive constant.

Functions u and v that are the real and imaginary part of a holomorphic function are called conjugateharmonic functions.

Theorem 4.9. If h : U → R is harmonic and f is a holomorphic function mapping U onto U, then thecomposition h ◦ f is harmonic on U.

3


5 Conformal maps

Definition 5.1. A conformal map is a map that preserves angles and orientation.

Theorem 5.2. A holomorphic function whose derivative has no zeros is a conformal map.

Theorem 5.3. Suppose a map f : U → C of a domain U ⊆ C is differentiable in the real sense andconformal. Then f is holomorphic.

4©• stereographic projection: conformal and circle preserving• Riemann sphere• extended complex plane: C= C∪{∞}, with topology inherited from S2 via stereographic projection• Mercator projection

6 Möbius transformations

Definition 6.1. A Möbius transformation of the complex plane is a function of the form

f (z) =az + bcz + d

, where ad − bc 6= 0.

• W.l.o.g., one may assume ad − bc = 1.• similarity transformations z 7→ az + b• inversion z 7→ 1

z (geometric interpretation)• inverse, composition

Theorem 6.2. The Möbius transformations form a group of maps C→ C, the Möbius groupM . Themap

SL(2,C)→M ,�

a bc d

�

7−→�

z 7→az + bcz + d

�

is a group homomorphism with kernel {+1,−1}. The Möbius group M is therefore isomorphic toSL(2,C)/{+1,−1}.

Theorem 6.3. Every Möbius transformation is either a similarity transformation or the composition ofa similarity transformation, the inversion z 7→ 1

z , and a similarity transformation.

Theorem 6.4. A Möbius transformation maps circles to circles. (Straight lines are considered as circlesthrough∞.)

5©• Warning: The center of a circle is in general not mapped to the center of the image circle.

Theorem 6.5. For three different points z1, z2, z3 in C and three different points w1, w2, w3, there is aunique Möbius transformation f with f (zk) = wk, k = 1,2, 3.

Example 6.6. The Möbius transformation

f (z) =z − iz + i

maps the real axis (incl. ∞) to the unit circle and the upper half plane H = {z | Im z > 0} to the unitdisk D = {z | |z|< 1}.

4


Definition 6.7. The cross-ratio of four different points z1, z2, z3, z4 in C is

cr(z1, z2, z3, z4) =(z1 − z2)(z3 − z4)(z2 − z3)(z4 − z1)

(1)

Equivalently, cr(z1, z2, z3, z4) is the image of z1 under the Möbius transformation that maps z2, z3, z4to 0, 1,∞, respectively.

• If one of the four points is ∞, the right hand side of equation (1) is extended continuously, bysensibly “cancelling infinities”. For example,

cr(z1, z2,∞, z4) =(z1 − z2)��

��:−1(∞− z4)

��(z2 −∞)(z4 − z1)

= −z1 − z2

z4 − z1.

Warning: Definitions with different permutations of the arguments are common in the literature.

Theorem 6.8. Four different points lie on a circle if and only if their cross-ratio is real.

Theorem 6.9. For every Möbius transformation and any four different points z1, z2, z3, z4,

cr�

f (z1), f (z2), f (z3), f (z4)�

= cr(z1, z2, z3, z4).

Theorem 6.10. (i) A Möbius transformations f maps the upper half plane

H = {z ∈ C | Im z > 0}

onto itself if and only if there are real coefficients a, b, c, d with ad−bc > 0 (or, equivalently, ad−bc = 1)such that f (z) = az+b

cz+d . 6©

(ii) A Möbius transformation f maps the unit disk D onto D if and only if one, and hence both, of thefollowing conditions are satisfied:

• There are a, b ∈ C such that |a|2 − |b|2 > 0 (or, equivalently, |a|2 − |b|2 = 1) and

f (z) =az + bbz + a

.

• There is a number φ ∈ R and a point z0 ∈ D such that

f (z) = eiφ z − z0

1− z0z.

5


7 Complex integration

• If h : [a, b]→ Rn is continuous, then the integral∫ b

a h(t) d t is defined component-wise as Riemannor Lebesgue integral.

Definition 7.1 (contour integral). Let U ⊆ C be an open subset, let f : U → C be a continuousfunction, and let γ : [a, b]→ U be continuously differentiable. Then

∫

γ

f (z) dz :=

∫ b

a

f (γ(t))γ′(t) d t

is called the [contour] integral of the function f along the curve γ. If γ is only piecewise continuouslydifferentiable, then contour integral is the sum of integrals, as defined above, over the sub-intervalswhere γ is differentiable.

• The contour integral is invariant under re-parameterization of the curve γ. More precisely, if φ :[c, d]→ [a, b] is continuously differentiable and φ(c) = a, φ(d) = b, then

∫

γ

f (z) dz =

∫

γ◦φf (z) dz.

If φ(c) = b, φ(d) = a, then the integral changes sign:∫

γ

f (z) dz = −∫

γ◦φf (z) dz.

• If f is bounded by C ∈ R (i.e., | f (z)| ≤ C for all z ∈ U) then

�

�

�

∫

γ

f (z) dz�

�

�≤ C length(γ).

7©Theorem 7.2 (Fundamental theorem of complex calculus). Let U ⊆ C be an open subset, and letf : U → C be a continuous function. Then the following conditions are equivalent:

(i) For all closed paths η in U,∫

η

f (z) dz = 0.

(ii) The integral of f along a path γ : [a, b]→ U depends only on the endpoints γ(a), γ(b).(iii) There is a holomorphic function F : U → C with F ′ = f .

If one, and hence all of these statements are true, then∫

γ

f (z) dz = F(γ(b))− F(γ(a)).

Example 7.3.∫

γzn dz, where γ : [0, 2π]→ C, γ(t) = rei t , r > 0.

• Notation: If γ is a circle around z0 with radius r, i.e.,

γ : [0, 2π]→ C, γ(t) = z0 + rei t ,

one writes∫

|z−z0|=r . . . dz for∫

γ. . . dz.

6


Theorem 7.4 (Integration of power series). Suppose the power series f (z) =∑∞

k=0 ak(z − z0)k hasradius of convergence R > 0. Then the power series F(z) =

∑∞k=0

akk+1 (z − z0)k+1 also has radius of

convergence R, and F ′ = f .8©

Theorem 7.5 (Differentiation of power series). A power series f (z) =∑∞

k=0 ak(z− z0)k with radius ofconvergence R> 0 is holomorphic on {z | |z−z0|< R} and the derivative is f ′(z) =

∑∞k=1 kak(z−z0)k−1,

which is a power series with the same radius of convergence R.

8 Vector analysis interpretation of the contour integral

Let U ∈ R2 = C be an open subset, let v = v1 + iv2 : U → C be a vector field, and let

f = v.

Then, for a piecewise C1 curve γ : [a, b]→ U ,∫

γ

f (z)dz =

∫ b

a

⟨v(γ(t)),γ′(t)⟩ d t︸︷︷︸

integral of v along γ

+i

∫ b

a

det�

v(γ(t)),γ′(t)�

d t︸︷︷︸

flow of v through γ

.

Now suppose γ is the boundary curve of a bounded domain G ∈ C . Then the classical integraltheorems of Gauss and Stokes (2D version) say

flow of v through γ=

∫

G

div(v) d x1 d x2, (Gauss)

integral of v along γ=

∫

G

rot(v) d x1 d x2, (2D Stokes)

wherediv(v) = ∂1v1 + ∂2v2, rot(v) = ∂1v2 − ∂2v1.

The Cauchy–Riemann-equations for f are equivalent to div v = rot v = 0.

Theorem 8.1 (Cauchy’s integral theorem, vector analysis version). Let f be holomorphic on U::::with

:::::::::continuous

::::::::derivative

:::f ′, and let the curve γ in U be the piecewise C1 boundary curve of a bounded

domain G ∈ C. Then∫

γ

f (z) dz = 0.

9 Cauchy’s integral theorem

Theorem 9.1 (Cauchy’s integral theorem for rectangles). Let f be holomorphic in U (by definition,this implies that U is open in C), let

Q = {z | Re z ∈ [a, b], Im z ∈ [c, d]}

be a closed rectangle contained in U, and let γ be a piecewise continuously differentiable parameterizationof the boundary curve of Q. Then

∫

γ

f (z) dz = 0.

7


Theorem 9.2 (Cauchy’s integral theorem for C1-images of rectangles). Let f be holomorphic on U, let 9©Q = [a, b] + i[c, d] be a closed rectangle in C, let γ be a piecewise C1 parameterization of the boundaryof Q, and let φ : Q→ C be a continuously differentiable function such that φ(Q) ⊂ U.Then

∫

φ◦γf (z) dz = 0.

• “Remember”: If W ⊆ Rm is open, then the derivative of a function φ : W → Rn is defined interms of approximating linear functions. The definition is extended to functions φ whose domainof definition W is not open by assuming that φ can be extended to a differentiable function on anopen neighborhood of W .

• Corollaries:� Cauchy’s integral theorem for triangles� Cauchy’s integral theorem for disks� Cauchy’s integral theorem for annuli [Kreisringe]� Cauchy’s integral theorem for nested non-concentric circles� Cauchy’s integral theorem for C1-homotopic curves (→ homework?)� Cauchy’s theorem for freely C1-homotopic closed curves (→ homework?)

10 Cauchy’s integral formula and the mean value theorem

Theorem 10.1 (Cauchy’s integral formula for a disk). Let f be holomorphic on a domain that containsthe closed disk {z ∈ C | |z − z0| ≤ r}. Then, for all points a in the interior of the disk,

f (a) =1

2πi

∫

|z−z0|=r

f (z)z − a

dz.

Theorem 10.2 (Mean value theorem). Let f be holomorphic on a domain that contains the closed disk 10©{z ∈ C | |z− z0| ≤ r}. Then the value of f at the center of the disk is the mean of the values on the circle|z − z0|= r, i.e.,

f (z0) =1

2π

∫ 2π

0

f (z0 + reiφ) dφ.

11 Power series theorem and some consequences

Theorem 11.1 (power series theorem). Let f be holomorphic on U and suppose U contains the opendisk Dz0,R with center z0 and radius R> 0. Then, for all z ∈ Dz0,R,

f (z) =∞∑

k=0

ak(z − z0)k

where

ak =1

2πi

∫

|z−z0|=r

f (z)(z − z0)k+1

dz

with arbitrary r ∈ (0, R).

8


Corollary 11.2. Holomorphic functions are arbitrarily often differentiable.

In particular, holomorphic functions are continuously differentiable, and the real inverse functiontheorem for continuously differentiable functions implies the complex inverse function theorem:

Corollary 11.3 (Inverse function theorem). Let f be holomorphic on U, and let z0 ∈ U with f ′(z0) 6= 0.Then there is an open neighborhood U0 ⊆ U of z0 such that f is injective on U0, f (U0) is open, and theinverse ( f |U0

)−1 is holomorphic on U0.

Corollary 11.4 (Cauchy integral formula for derivatives). Let f be holomorphic on a domain thatcontains the closed disk {z ∈ C | |z − z0| ≤ r}. Then, for all points w in the interior of the disk,

f (k)(w) =k!

2πi

∫

|z−z0|=r

f (z)(z −w)k+1

dz.

Corollary 11.5 (Estimate for Taylor coefficients). Let f (z) =∑∞

k=0 ak(z − z0)k be a power series withradius of convergence R ∈ (0,∞]. Suppose 0 < r < R and | f (z)| ≤ M for all z with |z − z0| ≤ r. Thenfor all k ∈ {0, 1,2, . . .}

|ak| ≤Mrk

.

Corollary 11.6 (Liouville’s theorem). A bounded entire function is constant.

Corollary 11.7 (Fundamental theorem of algebra). Every non-constant complex polynomial has a zero.

• Using polynomial division, one deduces that the number of zeros (counting multiplicities) of anon-zero polynomial is equal to the degree.

12 Zeros of holomorphic functions 11©

Definition 12.1 (Order of a zero). A holomorphic function f has a zero of order n in a, if

f (k)(a) = 0 if 0< k < n,

f (n)(a) 6= 0.

Proposition 12.2 (Zero of infinite order). If f is holomorphic on a domain U, and f (k)(a) = 0 forsome a ∈ U and for all k ∈ Z≥0, then f = 0.

Proposition 12.3. A holomorphic function f on U has a zero of order n at a ∈ U if and only if

f (z) = (z − a)n g(z)

for some holomorphic function g on U with g(a) 6= 0.

Theorem 12.4. If f has a zero of order n at z0, then there is an open neighborhood U0 of z0 and anholomorphic function h on U0 with a simple zero at z0 such that

f (z) = h(z)n.

Theorem 12.5 (Behavior of a holomorphic function near a zero). If f has a zero of order n at z0 andif ε > 0 is small enough, then there is an open neighborhood Uε 3 z0 such that

• f maps Uε onto the ε-disk {w | |w|< ε},• f attains every value w with 0< |w|< ε exactly n times in Uε,• f attains the value 0 in U0 only at z0.

Corollary 12.6. Zeros of finite order of a holomorphic function are isolated.

9


13 Identity theorem and invariance of domain

Theorem 13.1 (Identity theorem). Let f and g be holomorphic functions on a domain U, and supposef (z) = g(z) for all z in a subset of U that has an accumulation point in U. Then f = g.

Theorem 13.2 (Invariance of domain). If f is holomorphic an not constant on a domain U, then f (U)is also a domain. 12©

Theorem 13.3 (Maximum principle). If f is holomorphic and not constant on a domain U, then thereal valued functions | f |, Re f , Im f do not attain a maximum.

Corollary 13.4 (Maximum principle, boundary version). If f is holomorphic on a domain U, and ifA⊆ U is a compact subset, then | f |, Re f , Im f attain their maximum on the boundary of A.

Theorem 13.5 (Schwartz’s Lemma). If f is a holomorphic function on the unit disk D with imagef (D) ⊆ D, and f (0) = 0. Then | f ′(0)| ≤ 1 and | f (z)| ≤ |z| for all z. If | f ′(0)| = 1 or | f (z0)| = |z0| forsome z0 6= 0, then f is rotation: f (z) = eiαz for some α ∈ R.

Theorem 13.6 (Automorphisms of disk and half-plane). (i) Suppose f is an injective holomorphicmap on the unit disk D with image f (D) = D. Then f is a Möbius transformation mapping D onto D.

(ii) Suppose f is an injective holomorphic map on the upper half-plane H with image f (H) = H. Thenf is a Möbius transformation mapping H onto H.

Definition 13.7. An injective holomorphic map with holomorphic inverse is called biholomorphic.

Theorem 13.8. An injective holomorphic map is biholomorphic.

14 Morera’s theorem and the reflection principle

Theorem 14.1 (Morera). Let f : U → C be a continuous function on an open subset U ⊆ C. If∫

γ

f (z) dz = 0

whenever γ is the boundary curve of a closed triangular region contained in U, then f is holomorphic.13©

Theorem 14.2 (Schwarz reflection principle). Let U ⊆ C be an open subset which is symmetric withrespect to reflection on the real axis, that is, z ∈ U ⇔ z ∈ U. Suppose the function

f : U ∩ {z | Im z ≥ 0} −→ C

is continuous, holomorphic on U ∩ {z | Im z > 0} and real valued on U ∩R. Then the function

f : U → C, f (z) =

¨

f (z) if Im z ≥ 0

f (z) if Im z < 0

is holomorphic.

10


15 Isolated singularities

Definition 15.1 (isolated singularity). A point z0 ∈ C is called an isolated singularity of a holomorphicfunction f on U ⊆ C if z0 6∈ U , but there is an ε > 0 such that

{z ∈ C |0< |z − z0|< ε} ⊆ U .

Definition 15.2 (removable singularity). An isolated singularity z0 of a holomorphic function f onU is called removable if there is a holomorphic function f on U ∪ {z0} with f |U = f .

Theorem 15.3 (Riemann’s theorem on removable singularities). If z0 is a removable singularity of theholomorphic function f , then the following statements are equivalent:

(i) z0 is a removable singularity.(ii) f is bounded in a neighborhood of z0. That is, there is an ε > 0 such that the restriction of f to

�

z ∈ C�

�0< |z − z0|< ε

is bounded.(iii) lim

z→z0

(z − z0) f (z) = 0

Definition and Theorem 15.4 (Three types of isolated singularities). Let z0 be an isolated singularityof f . Then exactly one of the following statements is true:

(i) f (z) is bounded in a neighborhood of z0 and z0 is a removable singularity.(ii) limz→z0

| f (z)| =∞. Then z0 is called a pole and (z − z0)m f (z) has a removable singularity forsome m ∈ Z>0. The smallest m with this property is called the order of the pole.

(iii) Neither (i) nor (ii) are true. Then z0 is called an essential singularity.

Examples 15.5. (i) f (z) = 11−z2

(ii) g(z) = 1sin z

(iii) h(z) = zsin z

(iv) q(z) = e1/z

Theorem 15.6 (Cassorati–Weierstrass). Let z0 be an essential singularity of f : U → C and let ε > 0 14©be small enough so that U0 = {z |0< |z − z0|< ε} ⊆ U. Then f (U0) is dense in C.

• In other words, the conclusion of Theorem 15.6 says the following: For every w ∈ C, every δ > 0,and every ε > 0, there is a z ∈ U with |z − z0|< ε such that | f (z)−w|< δ.

• In fact, a stronger statement is true:

Theorem 15.7 (Great Picard’s theorem). In every neighborhood of an essential singularity, a holomor-phic function attains every value in C with at most one exception. [no proof given]

Definition 15.8. Let U ⊆ C be an open subset. A function is called holomorphic on U up to isolatedsingularities if f is holomorphic on U \ S for some subset S ⊆ U and all points of S are isolatedsingularities. If all the isolated singularities are poles, then f is called meromorphic.

Proposition 15.9. If f and g are holomorphic on U and g is not constantly zero, then fg is meromorphic

on U (after all removable singularities are removed).

Remark 15.10. The set of meromorphic functions on an open subset U ⊆ C is denoted by M (U).If + and · are defined on M (U) by first adding/multiplying functions on the intersection of theirdomains, then removing all removable singularities, thenM (U) is a ring with the constant functions0 and 1 as 0-element and 1-element, respectively. If U is a domain, thenM (U) is a field.

11


16 Laurent series

A Laurent series is a series of the form∞∑

k=−∞

ak(z − z0)k. (2)

More precisely, a Laurent series is a pair of two series∞∑

k=1

a−k(z − z0)−k

︸︷︷︸

principal part

and∞∑

k=0

ak(z − z0)k

︸︷︷︸

holomorphic part

.

The holomorphic part is a normal power series with some radius of convergence 0 ≤ R ≤∞. Theprincipal part is a power series in 1

z−z0. Let 1

r be its radius of convergence. If 0 ≤ r < R ≤∞, thenboth series converge on the annulus

A= {z ∈ C | r < |z − z0|< R}. (3)

In this case the Laurent series is said to converge on A, and∑∞

k=−∞ ak(z − z0)k denotes the sum ofthe limits of both series, the principal and the holomorphic part. Both parts converge absolutely onA and uniformly on compact subsets of A.

Laurent series can be differentiated term by term. One can take antiderivatives term by term as longas a−1 = 0. (Both statements can be seen by substituting ζ= 1

z−z0in the principal part.)

Theorem 16.1 (Uniqueness of Laurent series). Suppose the Laurent series f (z) =∑∞

k=−∞(z − z0)k

converges on the annulus (3). Then for all k ∈ Z and any ρ ∈ (r, R),

ak =1

2πi

∫

|z−z0|=ρ

f (z)(z − z0)k+1

dz. (4)

In particular, the integral does not depend on ρ.

Lemma 16.2 (Cauchy formula for annuli). Let A= {z | r < |z− z0|< R} 6= ; and let f be holomorphic 15©on a domain containing A. Then, for z ∈ A and ρ1, ρ2 satisfying r < ρ1 < |z − z0|< ρ2 < R,

f (z) =1

2πi

∫

|u−z0|=ρ2

f (u)u− z

du−1

2πi

∫

|u−z0|=ρ1

f (u)u− z

du .

Theorem 16.3 (Laurent series theorem). Let A= {z | r < |z − z0| < R} 6= ; and let f be holomorphicon a domain containing A. Then, for all z ∈ A,

f (z) =∞∑

k−∞

ak(z − z0)k,

where ak defined by (4) with arbitrary ρ ∈ (r, R).

• Suppose f has an isolated singularity at z0. Then f has a convergent Laurent series on 0< |z−z0|<R for some R> 0. The principal part of this Laurent series is also called the principal part of f in z0.

• There are three possibilities:� ak = 0 for all k < 0. Then z0 is removable.� ak 6= 0 some but only finitely many k < 0. Then the Laurent series is

∑∞k=−n ak(z − z0)k with

a−n6=0, and z0 is a pole of order n.� ak 6= 0 for infinitely many k < 0. Then z0 is an essential singularity.

Example 16.4. Laurent series of f (z) = 11−z in 0< |z − 1|, |z|< 1 and in 1< |z|.

12


17 Analytic continuation 16©

Definition 17.1. A function element is a pair ( f , U) of a domain U ⊆ C and a holomorphic function fon U . Two function elements ( f , U) and (g, V ) are called equivalent at a point a ∈ C, written

( f , U) a∼ (g, V ),

if a ∈ U ∩ V and f (z) = g(z) in a neighborhood of a.

• The relationa∼ is an equivalence relation on the set of function elements.

Definition 17.2 (analytic continuation). A function element (g, V ) is called an analytic continuationof a function element ( f , U), if there is a sequence of function elements

( f , U) = ( f0, U0), ( f1, U1), . . . , ( fn, Un) = (g, V )

and a sequence of points a0, . . . , an−1, such that

( fk, Uk)ak∼ ( fk+1, Uk+1)

for all k ∈ {0, . . . , n− 1}.

Proposition 17.3 (algebraic and differential equations). Let ak and b be entire functions for k ∈{0, . . . , n}.

(i) If a function element ( f , U) satisfies the equation

an(z)( f (z))n + . . .+ a1(z) f (z) + a0(z) = 0

for all z ∈ U, then any analytic continuation (g, V ) also satisfies

an(z)(g(z))n + . . .+ a1(z)g(z) + a0(z) = 0

for all z ∈ V .

(ii) If a function element ( f , U) satisfies the differential equation

an(z) f(n)(z) + . . .+ a1(z) f

′(z) + a0(z) f (z) = b(z)

on U, then any analytic continuation (g, V ) satisfies

an(z)g(n)(z) + . . .+ a1(z)g

′(z) + a0(z)g(z) = b(z)

on V .

Examples 17.4. • f (z)2 = z• f ′(z) = 1

z

Definition 17.5 (analytic continuation along a curve). Let c : [0,1]→ C be a continuous curve. Ananalytic continuation along c is a family ( ft , Ut)t∈[0,1] of function elements, such that

(i) For all t ∈ [0, 1], c(t) ∈ Ut .(ii) For every t ∈ [0, 1], there is a δ > 0 such that for all s ∈ [0,1] with |s− t|< δ

( ft , Ut)c(s)∼ ( fs, Us).

13


In this case, one says that ( f1, U1) is an analytic continuation of ( f0, U0) along the curve c.

• A function element ( f1, U1) is an analytic continuation of ( f0, U0) (by Definition 17.2) if and onlyif there is a curve c such that ( f1, U1) is an analytic continuation of ( f0, U0) along c (by Defini-tion 17.5).

• It is obvious how to define analytic continuation along curves with parameter interval [a, b] otherthan [0, 1].

Example 17.6 (trivial continuation). Let f be holomorphic on a domain U , and let c : [0,1]→ U bea curve in U . Then ( ft , Ut) = ( f , U) is an analytic continuation along c.

Example 17.7 (logarithm). Let c : [0,1]→ C \ {0} be a C1-curve with c(0) = 1. For each t ∈ [0, 1],there is an antiderivative Lt of z 7→ 1

z , defined on the open disk

Dt =�

z�

� |z − c(t)|< |c(t)|

,

and with

Lt(c(t)) =

∫

c|[0,t]

1z

dz.

In particular, L0 is the principal value logarithm function restricted to D0.

The family (Lt , Dt)t∈[0,1] is an analytic continuation along c.

Lemma 17.8 (continuation and power series). Let ( ft , Ut) be an analytic continuation along a curve c :[0,1]→ C. For each t ∈ [0, 1], let ft be the holomorphic function on Dt represented by the power seriesexpansion of ft around c(t), where Dt is the disk of convergence of the power series. Then ( ft , Dt)t∈[0,1]

is also an analytic continuation along c, and clearly ( ft , Ut)c(t)∼ ( ft , Dt) for all t ∈ [0, 1].

Theorem 17.9 (Uniqueness). Let ( ft , Ut) and (gt , Vt) be two analytic continuations along a curve 17©c : [0,1]→ C. Then

( f0, U0)c(0)∼ (g0, V0) =⇒ ( f1, U1)

c(1)∼ (g1, V1).

Lemma 17.10 (Lebesgue number). Let (U j) j∈J be an open cover of a compact metric space X . Thenthere is an ε > 0 such that for every subset A⊆ X with diam(A)< ε, there is a j ∈ J with A⊆ U j .

Lemma 17.11. Let ( ft , Ut) be an analytic continuation along a curve c : [0, 1] → C. Then there is asubdivision

0= t0 < t1 < . . .< tn = 1

of [0, 1] and a sequence of function elements (gk, Dk)1≤k≤n−1, where Dk are disks, such that

( ft , Ut)c(t)∼ (gk, Dk) for all t ∈ [tk−1, tk+1].

Lemma 17.12. Let c : [0,1]→ C be a continuous curve, and let U0 be an open neighborhood of c(0). Afunction element ( f0, U0) can be continued analytically along c if and only if the derived function element( f ′(0), U0) can be continued analytically along c.

Definition 17.13 (Integration along continuous curves). Let f be a holomorphic function on U andlet c : [0, 1]→ U be a continuous curve. We define

∫

c

f (z) dz := F1(c(1))− F0(c(0)), (5)

where F0 and F1 are defined as follows: For a small disk U0 around c(0), the function f |U0has an

antiderivative F0. Since ( f , U0) = (F ′, U0) can be continued (trivially) along c, so can (F0, U0). Let(Ft , Ut)t∈[0,1] be such an analytic continuation.

14


Proposition 17.14. If c is piecewise C1, the integral of Definition 17.13 has the same value as theintegral of Definition 7.1.

Theorem 17.15 (Cauchy’s integral theorem for continuous images of rectangles). Let f be holomor- 18©phic on U, let Q = [0,1] × [0, 1], let φ : Q → U be a continuous function, let γ : [0,1] → R2 be aparametrization of the boundary ∂Q, and let γ= φ ◦ γ. Then

∫

γ

f (z) dz = 0.

18 Homotopy1

In the following U ⊆ C is an open subset. But most definitions and theorems remain valid if U is anytopological space.

Definition 18.1 (homotopic curves). Two curves c0, c1 : [0, 1]→ U with

c0(0) = c1(0), c0(1) = c1(1)

are called homotopic [in U] if there exists a homotopy between them, that is, a continuous map

H : [0, 1]× [0,1]→ U

(t,τ) 7→ H(t,τ)

such thatH(·, 0) = c0, H(·, 1) = c1

andH(0,τ) = c0(0) = c1(0), H(1,τ) = c0(1) = c1(1).

A closed curve c : [0,1]→ U is called null-homotopic [in U] if it is homotopic to the constant curvec1(t) = c(0).

Definition 18.2 (concatenation of curves). For continuous curves c1, c2, c : [0, 1]→ U with

c1(1) = c2(0),

define the concatenation c1c2 : [0, 1]→ U by

c1c2(t) =

¨

c1(2t) for 0≤ t ≤ 12

c2(2t − 1) for 12 ≤ t ≤ 1

and the backward traced curve cinv : [0, 1]→ U by

cinv(t) = c(1− t).

Lemma 18.3. Let c : [0,1]→ U be a continuous curve, and let φ : [0,1]→ [0,1] be a continuous mapwith

φ(0) = 0, φ(1) = 1.

Then1This section is essentially copied verbatim from Dirk Ferus’ notes, where you can also find the proofs.

15


(i) ccinv is null-homotopic.(ii) c ◦φ is homotopic to c.

Lemma 18.4. Let c1, c2, c3 : [0,1]→ U be continuous curves with

c1(1) = c2(0), c2(1) = c3(0).

Then (c1c2)c3 and c1(c2c3) are homotopic in U.

Theorem and Definition 18.5 (fundamental group). Let U ∈ C be an open subset, and let z0 ∈ U.In the set of all closed curves c : [0, 1] → U with c(0) = c(1) = z0 (the set of loops at z0), homotopydefines an equivalence relation. The concatenation of curves defines a multiplication on the set π1(U , z0)of equivalence classes, which turns this set into a group. It is called the fundamental group of U withbase point z0. The neutral element of the fundamental group is the class of the constant curve t 7→ z0,and the inverse element of the class a curve c is the class of cinf. The class of a curve c is denoted by[c] ∈ π1(U , z0). So we have

[c1][c2] = [c1c2], [c]−1 = [cinv].

• The fundamental group π1(U , z0) depends on the base point. But if U is a domain (or any otherpath-connected topological space) and z1 ∈ U , then there is a curve γ : [0, 1]→ U with γ(0) = z0and γ(1) = z1. The map

π1(U , z0) 3 [c] 7−→ [γcγinv] ∈ π1(U , z1)

defines an group isomorphism π1(U , z0) −→ π1(U , z1).

Theorem and Definition 18.6 (simply connected). Let U be a domain in C (or a path-connected 19©topological space), and U 6= ;. Then the following statements are equivalent:

(i) Every closed curve c : [0,1]→ U is null-homotopic in U.(ii) π1(U , z0) = {1} for all z0 ∈ U.

(iii) π1(U , z0) = {1} for at least one z0 ∈ U.(iv) Any two curves c0, c1 : [0,1]→ U with c0(0) = c1(0), c0(1) = c1(1) are homotopic in U.

If one and hence all of these statements are true, then U is called simply connected.

Examples 18.7. • Convex sets are simply connected.• Star-shaped sets are simply connected.• The set C \R≤0 is not convex, but star-shaped.• The punctured plane C∗ := C \ {0} is not simply connected, and one can show that π1(C∗, 1)∼= Z.• The fundamental group of the twice punctured plane C \ {0, 1} is not abelian. It is isomorphic to

the free group with two generators.

19 The monodromy theorem

Theorem 19.1 (monodromy theorem). Let G ∈ C be a domain and z0 ∈ G. Suppose a function element( f , U) with z0 ∈ U can be continued analytically along any curve c : [0,1]→ U beginning in z0. If c0and c1 are two homotopic curves in G beginning in z0 and ending in z1, and if ( f0,t , U0,t)t and ( f1,t , U1,t)tare analytic continuations of ( f , U) along c0 and c1, respectively, then

( f0,1, U0,1)z1∼ ( f1,1, U1,1).

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20 Homology of curves2 20©

Let U ⊆ C be an open subset.

• A 0-chain in U is a formal suma1z1 ⊕ a2z2 ⊕ . . .⊕ anzn

of points zk ∈ U with coefficients ak ∈ Z.• The abelian group of 0-chains in U is denoted by C0(U).• A 1-chain in U is a formal sum

a1c1 ⊕ a2c2 ⊕ . . .⊕ ancn (6)

of curves ck : [0,1]→ U with coefficients ak ∈ Z.• The abelian group of 1-chains in U is denoted by C1(U).• For a curve c : [0,1]→ U , let

∂ c = c(1) c(0) ∈ C0(U).

For a 1-chain (6), let

∂�

n⊕

k=1

akck

�

=n⊕

k=1

ak ∂ ck

=�

n⊕

k=1

akck(1)�

�

n⊕

k=1

akck(0)�

.

This defines a group homomorphism ∂ : C1→ C0 called the boundary operator.• A 1-chain is called closed or a 2-cycle if ∂ c = 0.• The 1-chain c =

⊕

k akck in U is closed if and only if for all p ∈ U∑

k:ck(1)=p

ak =∑

k:ck(0)=p

ak

• Let f be a holomorphic function on U . For a 1-chain

c =n⊕

k=1

akck

in U define∫

c

f (z) dz =n∑

k=1

∫

ck

f (z) dz.

For a 0-chain⊕n

k=1 bkzk, define

∫

⊕nk=1 bkzk

f =n∑

k=1

ak f (zk).

Then∫

c

f ′(z) dz =

∫

∂ c

f .

2There are different homology theories. The one we consider here is called “singular homology with integer coefficients”.

17


• Let ∆= {(t1, t2) ∈ R2 | t1 ≥ 0, t2 ≥ 0, t1 + t2 ≤ 1}. A singular triangle in U is a continuous map

τ :∆→ U .

A 2-chain in U is a formal suma1τ1 ⊕ . . .⊕ anτn

of singular triangles τk with coefficients ak ∈ Z.• The abelian group of 2-chains in U is denoted by C2(U).• For a singular triangle τ :∆→ U , let

∂ τ= c1 c2 ⊕ c3 ∈ C1(U),

where ck : [0, 1]→ U ,

c1(t) = τ(t, 0), c2(t) = τ(0, t), c3(t) = τ(1− t, t).

For a 2-chain

τ=n⊕

k=1

akτk,

let

∂ τ=n⊕

k=1

ak ∂ τk.

This defines a group homomorphism ∂ : C2(U)→ C1(U), called the boundary operator.• A 1-chain c is called 0-homologous if there is a 2-chain τ with ∂ τ= c.• Two 1-chains c1 and c2 are called homologous, if c1 c2 is 0-homologous, that is, if there is a

2-chain τ with c1 = c2 ⊕τ.• A 0-homologous 1-chain is closed. The converse is not generally true: A closed 1-chain may not

be 0-homologous.• A constant curve c(t) = z0 ∈ U is 0-homologous. [exercise]• c ⊕ cinv is 0-homologous. [exercise]• The concatenation c1c2 of two curves with c0(1) = c1(0) is homologous to c1 ⊕ c2.• If c1 and c2 are loops at z0 in U then the concatenation c1c2cinv

1 cinv2 is null-homologous in U . [exer-

cise]• If c1 and c2 are homotopic in U then c1 and c2 are homologous in U . [exercise]• The converse is not generally true: A 0-homologous curve may not be null-homotopic.• If the closed curves c1 and c2 are freely homotopic in U , then c1 and c2 are homologous in U .[exercise]

This is the final version of Cauchy’s integral theorem: 21©

Theorem 20.1 (Cauchy’s integral theorem, homology version). If f is holomorphic on U and the curvec : I → U is 0-homologous in U, then

∫

c

f (z) dz = 0.

21 Winding number

Definition 21.1. The support a 1-chain c =⊕

k akck is

|c|=⋃

k:ak 6=0

ck([0,1]),

18


and the support a 2-chain τ=⊕

k akτk is

|τ|=⋃

k:ak 6=0

τk(∆).

Definition 21.2 (winding number). The winding number of a 1-cycle c around a point z0 6∈ |c|, alsocalled the index of z0 with respect to c, is defined as

indc(z0) =1

2πi

∫

c

dzz − z0

.

Example 21.3. For c(t) = a+ re2πint on [0, 1],

indc(a) = n and indc(z) = 0 if |z0 − a|> r.

Theorem 21.4. (i) The winding number indc(z0) of 1-cycle is an integer.

(ii) If c is a 1-cycle and z0 and z1 are contained in the same connected component of C \ |c|, then

indc(z0) = indc(z1).

(iii) If two 1-cycles c and c are homologous in C \ {z0}, then

indc(z0) = indc(z0).

• crossing rule for winding numbers

Theorem 21.5 (Artin’s homology criterion). A 1-cycle c is 0-homologous in an open subset U ⊆ C if 22©and only if indc a = 0 for every a ∈ C \ U.

A proof can be found in Dirk Ferus’ lecture notes.

Corollary 21.6 (converse of Cauchy’s integral theorem). If c is not 0-homologous in U, then there isa holomorphic function f on U such that

∫

c

f (z) dz 6= 0.

Definition 21.7 (simply bounding cycle). Let B ∈ C be a compact subset. A 1-cycle bounds B simplyif

(i) |c| ⊆ ∂ B,(ii) indc z = 1 if z is an interior point of B,

(iii) indc z = 0 if z ∈ C \ B.

Examples 21.8. (i)–(iv) disk, rectangle, annulus, Jordan domain(v) If the 1-cycle c bounds the compact set B simply, and if K1, . . . , Kn are disjoint closed disks in the

interior B◦ of B, then

B \n⋃

j=1

K◦j

is simply bounded byc ∂ K1 . . . ∂ Kn.

19


Theorem 21.9 (winding number version of Cauchy’s integral formula). Let f be holomorphic on Uand c be a 0-homologous 1-cycle in U. Then for every z0 ∈ U \ |c|,

indc(z0) f (z0) =1

2πi

∫

c

f (z)z − z0

dz.

In particular, if c bounds the compact set B ⊆ U simply, and z0 ∈ B◦, then

f (z0) =1

2π

∫

c

f (z)z − z0

dz.

22 The residue theorem

Definition 22.1 (residue). If f has an isolated singularity at z0, then f is holomorphic on

A= {z |0< |z − z0|< R}

for some R > 0, and f can be expanded into a Laurent series f (z) =∑∞

k=−∞ ak(z − z0)k on A. Theresidue of f at z0 is

Res( f , z0) = a−1.

• By the Laurent series theorem, if 0< r < R,

Res( f , z0) =1

2πi

∫

|z−z0|=r

f (z) dz.

Theorem 22.2 (residue theorem v1). Let f be holomorphic in a domain U except for a set S of isolatedsingularities. If a 1-cycle bounds a compact set A⊆ U simply, and S ∩ ∂ B = ;, then

12πi

∫

c

f (z) dz =∑

z∈S∩B

Res( f , z).

Theorem 22.3 (residue theorem v2). If f is holomorphic on a domain U except for a set S of isolatedsingularities, and if c is a cycle in U \ S that is 0-homologous in U, then the set

{z ∈ S | indc(z) 6= 0}

is finite and1

2πi

∫

c

f (z) dz =∑

z∈S

indc(z)Res( f , z).

Example 22.4. If f and g are holomorphic in a neighborhood of z0 and g has a simple zero at z0, 23©then

Res(fg

, z0) =f (z0)g ′(z0)

.

Example 22.5. If f has a zero of order m or a pole of order −m at z0, then

Res� f ′

f, z0

�

= m.

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Theorem 22.6 (counting zeros and poles with an integral, and with a winding number). Let f bemeromorphic on U and let B ⊆ U be a compact region simply bounded by the 1-cycle c. Assume that fhas no poles or zeros on the boundary of B. Let p1, . . . , pm be the poles of f in B, and let a1, . . . , am ∈ Z>0be their pole orders. Let q1, . . . , qn be the zeros of f in B, and let b1, . . . , bn ∈ Z>0 be their orders. Then

12πi

∫

f ′(z)f (z)

dz =n∑

k=1

bk −m∑

k=1

ak = ind f ◦c(0).

• Used in the proof of Theorem 22.6:

Lemma 22.7 (substitution rule for integrals along continuous curves). If c : [a, b]→ C is a continuouscurve, φ is holomorphic in a neighborhood of |c|, and f is holomorphic in a neighborhood of |φ ◦ c| =φ(|c|), then

∫

φ◦cf (z) dz =

∫

c

( f ◦φ)(z)φ′(z) dz.

Theorem 22.8 (Rouché). Let f and g be holomorphic on U, and let B ⊆ U be a compact region simplybounded by a continuous curve c. Suppose

| f (z)− g(z)|< | f (z)| for all z ∈ ∂ B.

Then f and g have the same number of zeros in B, counted with multiplicities (i.e., order).

Example 22.9.

∫ ∞

−∞

cos xx2 + 1

d x =π

e

21

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