CUENTA DE PRODUCCIÓN

Before we study the main results of holomorphic functions in several complex variables, we listen the basic facts in one complex variable.

3.1 Complex Analysis in One Variable

3.1.1. Definition (Holomorphic Functions):

Let

U

⊆

C

be an open set and

f : U→ C

a function.

f is called complex differentiable in

z

₀ ∈

U

if the limit

L := lim

z→z0

f(z)−f(z0) z−z0

exists. In this case

L

is called the derivative of

f

at

z

₀. Notation:

f

′

(z

₀

) =

df(z

0

)

dz

=

z

lim

→z0

f(z) − f(z

₀

)

z − z

₀

If

f

is complex differentiable at every point

z

₀ in

U

, we say

f

is holomor- phic on

U

. We say

f

is holomorphic at the point

z

₀ ∈

U

, if it is

holomorphic on some neighbourhood

U

₀ of

z

₀.

3.1.2. Remark:

Equivalent to the holomorphy of

f

are:

(1) Characterising via the Cauchy - Riemann differential equations:

f

has continuous partial derivatives with respect to

x

and

y

at each point in

U

and satisfies

∂u

∂x

=

∂v

∂y

or shorter

u

x

= v

y

∂u

∂y

= −

∂v

∂x

or shorter

u

y

= − v

y (3.1.)

the Cauchy - Riemann differential equations, whereupon we write for

f := u + i v

.

This is, by the Wirtinger calculus equivalent to

∂f = 0

, with

∂f =

_∂z∂f

:=

1₂ (_∂x∂f

+ i

_∂y∂f). See [Kra08] for more on this matter. (2) Characterising via power series resp. Taylor series:

Locally,

f

is representable by a convergent power series which is equal to its Taylor series.

3.1 Complex Analysis in One Variable

A power series

∑

∞

n=0

a

n

(z − z

0

)

n around a fixed

z

0 ∈

C

is defined to

be the limit of its partial sums

S

_N

(z) =

N

∑

n=0

a

n

(z−z

₀

)

n,

(a

n∈

C, n

∈

N

₀

)

.

Any given power series has a disk/ball of convergence and the radius of convergence is given, by the Cauchy - Hadamard theorem, as

r : =

₍

1

lim sup

n→∞

|a

n

|

)1 n

.

(3.2.)

(3) Characterising via the complex line integral:

Cauchy - Integral Theorem :

If

f

is a holomorphic function on an open disc

W

in the complex plane, and if

γ : [a, b]

→ W

is a

C

1-curve in

W

with

γ(a) = γ(b)

, then

ˆ

γ

f(z) dz = 0.

The expression on the left -hand side is called the complex line integral of

f

along

γ

and is defined as:

ˆ

γ

f(z) dz : =

b

ˆ

a

f(γ(t))

dγ

dt(t) dt

The necessity of this condition is just the basic form of the Cauchy - Integral - Theorem, while Morera's Theorem shows that it is even sufficient.

Morera's Theorem: Let

U

⊆

C

be open,

f

continuous in

U

and all trian- gles

∆

⊆

U

satisfy

ˆ

b ∆

f(z) dz = 0

. Then

f

is holomorphic on

U

.

3.1.3. Theorem (Cauchy Integral Formula (Basic Form)):

Let

Ω

⊆

C

be a domain and suppose that

(r > 0), D

r

(z

₀

)

⊆

Ω

. Let

γ : [0, 1]→ C

be the

C

1 parametrization

γ(t) = z

₀

+ r cos(2πt) + ir sin(2πt)

3.1 Complex Analysis in One Variable

and

f

holomorphic in

Ω

. Then, for each

z

∈

D(z

₀

, r)

f(z) =

1

2πi

ˆ

γ

f(ξ)

ξ − z

dξ

(3.3.)

3.1.4. Remark:

For more general versions of the Cauchy Integral formula see [Kra08].

3.1.5. Theorem (Taylor series expansion):

Let

U

⊆

C

open and

a

∈

U, R > 0

such that

D

_R

(a)

⊆

U

and

f

holomorphic in

U

. Then

f(z) =

∞

∑

n=0

a

n

(z − a)

n

,

(3.4.)

where the sum converges uniformly on all compact subsets of

D

_R

(a)

and the following identity holds:

a

n

=

f

(n)

_(a)

n!

, n

∈

N

0 (3.5.)

a

n are called Taylor coefficients of

f

by expansion at

a

.

3.1.6. Theorem:

Let

f

be holomorphic in

U

, then

f

(n) is holomorphic in

U

∀

n

∈

N

₀.

f

is therefore infinitely complex differentiable.

3.1.7. Theorem:

Assumption as in 3.1.5.,

γ(t) = a + re

it

, r < R, t, t

∈

[0, 2π]

. Then

f

(n)

(a) =

n!

2πi

ˆ

γ

f(ξ)

(ξ − a)

(n+1)

dξ (n

∈

N

0

)

(3.6.)

3.1.8. Remark:

For proofs of the previous theorems see [Kra08] resp. [Has05].

3.1.9. Theorem (Weierstraß' Convergence Theorem):

Let

U

⊆

C

be an open subset,

f

n holomorphic on

U, n

∈

N

₀. If the se-

3.1 Complex Analysis in One Variable

tion

f

, i.e.

f

n

→ f

uniformly on

K

, then

f

is holomorphic on

U

.

Furthermore, ∀

k

∈

N

₀

: f

(k)_n converges uniformly on compact sets to

f

(k). In addition, if

f

n ∈

H(U)

is a Cauchy sequence in terms of uniformly con-

vergence on compact subsets of

U

,i.e.

∀

K

⊆

U,

compact

,

∀

ε > 0

∃

N

_ε,K

> 0

such that

sup

z∈K

|f

n

− f

m

| < ε

∀

n, m > N

_ε,K

,

∀

z

∈

K,

then there exists

f

∈

H(U)

such that

f

n n→∞

−→ f

uniformly on compact sets

of

Ω

.

H(U)

is complete in terms of uniformly convergence on compact subsets of

U

.

3.1.10. Remark (The Space of Holomorphic Functions):

Let

U

⊆

C

be an open set. The vector space of all holomorphic func- tions on

U

, denoted by

H(U)

, can be turned into a complete topological vector space. The topology can be described by a metric and coincides with the topology of uniform convergence on compact subsets of

U

:

It is known that there exits a sequence of compact subsets

K

n ⊆

U

such

that

K

n ⊆

K

_n+1, ∞_∪ n=1

K

n

= U

and such that if

K

is any compact subset of

U

, then

K

⊆

K

n for some n.

We now define a sequence of semi - norms by

∥

f

∥_n

: =

sup

z∈Kn

|f(z)| , f

∈

H(U), n

∈

N

and it yields that ∥

f

∥_n ≤ ∥

f

∥_n+1. Now we can define the following

d(f, g) : =

∞

∑

n=1 (

1

2

)_n ∥

f − g

∥_n

1 +

∥

f − g

∥_n

(f, g

∈

H(U))

It can be shown that

d

is a metric, that the topology generated by

d

is independent of the choice of

K

n and that convergence in

d

is the same

as uniform convergence on compact subsets.

In document Departamento Administrativo Nacional de Estadística - Ministerio de Cultura (página 56-61)