aarjaneiro/planetmath huggingface dataset:
example of analytic continuation

example of analytic continuation

The

function

defined by

f

⁢

(

z

)

:=

∑

n

=

0

∞

z

n

=

1

+

z

+

z

2

+

…

is, as a sum of power series,

analytic

in the disc of convergence

D

=

{

z

∈

ℂ

⋮

|

z

|

<

1

}

.  The function

g

⁢

(

z

)

:=

1

1

-

i

⁢

∑

n

=

0

∞

(

z

-

i

1

-

i

)

n

=

1

1

-

i

+

z

-

i

(

1

-

i

)

2

+

(

z

-

i

)

2

(

1

-

i

)

3

+

…

similarly is analytic in the bigger disc

E

=

{

z

∈

ℂ

⋮

|

z

-

i

|

<

2

}

.  But we have

f

⁢

(

z

)

=

1

1

-

z

in

⁢

D

,

g

⁢

(

z

)

=

1

1

-

i

⋅

1

1

-

z

-

i

1

-

i

=

1

1

-

z

in

⁢

E

;

thus

f

⁢

(

z

)

and

g

⁢

(

z

)

coincide in the

intersection

domain

D

∩

E

.  So we can say that

g

⁢

(

z

)

is the

analytic continuation

of

f

⁢

(

z

)

to the domain

E

∖

D

.  It is clear that

1

1

-

z

is the analytic continuation of

f

⁢

(

z

)

to the domain

ℂ

∖

{

1

}

.

Title

example of analytic continuation

Canonical name

ExampleOfAnalyticContinuation

Date of creation

2013-03-22 16:52:06

Last modified on

2013-03-22 16:52:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Example

Classification

msc 30B40

Classification

msc 30A99

Related topic

RadiusOfConvergence

Related topic

GeometricSeries

Related topic

SetDifference

Generated on Fri Feb 9 20:16:13 2018 by

LaTeXML
proof of Ruffa’s formula for continuous functions

proof of Ruffa’s formula for continuous functions

Define

s

n

to be the following sum:

s

n

=

∑

m

=

1

2

n

-

1

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

Making the substitution

m

′

=

2

⁢

m

and using the fact that

1

+

(

-

1

)

m

′

=

0

when

m

′

is odd to express the sum over even values of

m

′

as a sum over all values of

m

′

, this becomes

s

n

=

∑

m

=

1

2

n

+

1

-

1

(

1

+

(

-

1

)

m

′

)

⁢

2

-

1

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

Subtracting this sum from

s

n

+

1

and simplifying gives

s

n

+

1

-

s

n

=

∑

m

=

1

2

n

+

1

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

Using the telescoping sum trick, we may write

s

k

=

∑

n

=

1

k

(

s

n

-

s

n

-

1

)

=

∑

n

=

1

k

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

To

complete

the proof, we must investigate the limit as

k

→

∞

. Since

f

is assumed

continuous

and the

interval

[

a

,

b

]

is

compact

,

f

is

uniformly continuous

. This means that, for every

ϵ

>

0

, there exists a

δ

>

0

such that

|

x

-

y

|

<

δ

implies

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

<

ϵ

. By the Archimedean property, there exists an integer

k

>

0

such that

2

k

⁢

δ

>

|

a

-

b

|

. Hence,

|

f

⁢

(

x

)

-

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

|

≤

ϵ

when

x

lies in the interval

[

a

+

(

m

-

1

)

⁢

(

b

-

a

)

/

2

n

,

a

+

(

m

+

1

)

⁢

(

b

-

a

)

/

2

n

]

. Thus,

(

a

-

b

)

⁢

s

k

+

|

a

-

b

|

⁢

ϵ

is a Darboux upper sum for the

integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

and

(

b

-

a

)

⁢

s

k

-

|

a

-

b

|

⁢

ϵ

is a Darboux lower sum. (Darboux’s definition of the integral may be thought of as a modern incarnation of the ancient method of exhaustion.) Hence

|

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

-

s

k

|

≤

|

a

-

b

|

⁢

ϵ

Taking the limit

ϵ

→

0

, we see that

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

∑

n

=

1

∞

A

n

=

(

b

-

a

)

⁢

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

Title

proof of Ruffa’s

formula

for

continuous functions

Canonical name

ProofOfRuffasFormulaForContinuousFunctions

Date of creation

2013-03-22 14:56:41

Last modified on

2013-03-22 14:56:41

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30B99

Classification

msc 26B15

Classification

msc 78A45

Generated on Fri Feb 9 20:16:18 2018 by

LaTeXML
theorems on complex function series

theorems on complex function series

Theorem 1.

If the

complex functions

f

1

,

f

2

,

f

3

,

…

are

continuous

on the path

γ

and the series

f

1

⁢

(

z

)

+

f

2

⁢

(

z

)

+

f

3

⁢

(

z

)

+

…

(1)

converges uniformly

on

γ

to the sum

function

F

, then one has

∫

γ

F

⁢

(

z

)

⁢

𝑑

z

=

∫

γ

f

1

⁢

(

z

)

⁢

𝑑

z

+

∫

γ

f

2

⁢

(

z

)

⁢

𝑑

z

+

∫

γ

f

3

⁢

(

z

)

⁢

𝑑

z

+

…

Theorem 2.

If the functions

f

1

,

f

2

,

f

3

,

…

are

holomorphic

in a domain

A

and the series (1) converges uniformly in every closed (

http://planetmath.org/ClosedSet

) disc of

A

, then also the sum function

F

of (1) is holomorphic in

A

and the equality

d

n

⁢

F

⁢

(

z

)

d

⁢

z

n

=

F

(

n

)

⁢

(

z

)

=

f

1

(

n

)

⁢

(

z

)

+

f

2

(

n

)

⁢

(

z

)

+

f

3

(

n

)

⁢

(

z

)

+

…

(2)

is true for every positive integer

n

in all points of

A

.  The series (2) converges uniformly in every

compact

subdomain of

A

.

Theorem 3.

If

f

⁢

(

z

)

is holomorphic in a domain

A

and

z

0

is a point of

A

, then one can expand

f

⁢

(

z

)

to a

power series

(the so-called

Taylor series

)

f

(

z

)

=

∑

n

=

0

∞

a

n

(

z

-

z

0

)

n

where

a

n

=

f

(

n

)

⁢

(

z

0

)

n

!

(

n

=

0

,

1

,

2

,

…

)

.

This is valid at least in the greatest disk

|

z

-

z

0

|

<

r

(

≦

∞

)

which contains points of

A

only.

Title

theorems on complex function series

Canonical name

TheoremsOnComplexFunctionSeries

Date of creation

2013-03-22 16:47:55

Last modified on

2013-03-22 16:47:55

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30B99

Classification

msc 40A30

Related topic

IdentityTheoremOfPowerSeries

Related topic

WeierstrassDoubleSeriesTheorem

Generated on Fri Feb 9 20:16:21 2018 by

LaTeXML
Hurwitz’s theorem

Hurwitz’s theorem

Define the ball at

z

0

of radius

r

as

B

⁢

(

z

0

,

r

)

=

{

z

∈

G

:

|

z

-

z

0

|

<

r

}

and

D

⁢

(

z

0

,

r

)

=

{

z

∈

G

:

|

z

-

z

0

|

≤

r

}

is the

closed ball

at

z

0

of radius

r

.

Theorem

(Hurwitz).

Let

G

⊂

C

be a region and suppose the

sequence

of

holomorphic functions

{

f

n

}

converges uniformly on

compact subsets

of

G

to a holomorphic function

f

. If

f

is not identically zero,

D

⁢

(

z

0

,

r

)

⊂

G

and

f

⁢

(

z

)

≠

0

for

z

such that

|

z

-

z

0

|

=

r

, then there exists an

N

such that for all

n

≥

N

f

and

f

n

have the same number of zeros in

B

⁢

(

z

0

,

r

)

.

What this

theorem

says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always

converges

to a holomorphic function but that’s another theorem altogether), the

function

is not identically zero and furthermore the function is not
zero on the

boundary

of some ball,
then

eventually

the functions of the sequence have the same number of zeros inside this ball as does the function.

Do note the requirement for

f

not being identically zero. For example the sequence

f

n

⁢

(

z

)

:=

1

n

converges uniformly on compact subsets to

f

⁢

(

z

)

:=

0

, but

f

n

have no zeros anywhere, while

f

is identically zero.

Also in general this result holds for

bounded

convex subsets

(

http://planetmath.org/ConvexSet

) but it is most
useful to for balls.

An immediate

consequence

of this theorem is this useful corollary.

Corollary

.

If

G

is a region and a sequence of holomorphic functions

{

f

n

}

converges
uniformly on compact subsets of

G

to a holomorphic function

f

, and furthermore if

f

n

never vanishes (is not zero for any point in

G

), then

f

is either identically zero or also never vanishes.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

Hurwitz’s theorem

Canonical name

HurwitzsTheorem

Date of creation

2013-03-22 14:17:55

Last modified on

2013-03-22 14:17:55

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C15

Related topic

CompositionAlgebraOverAlgebaicallyClosedFields

Related topic

CompositionAlgebrasOverMathbbR

Related topic

CompositionAlgebrasOverFiniteFields

Related topic

CompositionAlgebrasOverMathbbQ

Generated on Fri Feb 9 20:16:26 2018 by

LaTeXML
Mason-Stothers theorem

Mason-Stothers theorem

Mason’s theorem is often described as the

polynomial

case of the (currently unproven)

ABC conjecture

.

Theorem 1

(Mason-Stothers).

Let

f

⁢

(

z

)

,

g

⁢

(

z

)

,

h

⁢

(

z

)

∈

C

⁢

[

z

]

be such that

f

⁢

(

z

)

+

g

⁢

(

z

)

=

h

⁢

(

z

)

for all

z

, and such that

f

,

g

, and

h

are pair-wise relatively prime. Denote the number of distinct roots of the product

f

⁢

g

⁢

h

⁢

(

z

)

by

N

. Then

max

⁡

deg

⁡

{

f

,

g

,

h

}

+

1

≤

N

.

Title

Mason-Stothers theorem

Canonical name

MasonStothersTheorem

Date of creation

2013-03-22 13:49:24

Last modified on

2013-03-22 13:49:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 30C15

Synonym

Mason’s Theorem

Related topic

PolynomialAnalogonForFermatsLastTheorem

Generated on Fri Feb 9 20:16:27 2018 by

LaTeXML
zeroes of analytic functions are isolated

zeroes of analytic functions are isolated

The zeroes of a non-constant

analytic function

on

ℂ

are isolated.
Let

f

be an analytic function
defined in some domain

D

⊂

ℂ

and
let

f

⁢

(

z

0

)

=

0

for some

z

0

∈

D

. Because

f

is analytic,
there is a

Taylor series

expansion for

f

around

z

0

which
converges on an

open disk

|

z

-

z

0

|

<

R

. Write it as

f

⁢

(

z

)

=

Σ

n

=

k

∞

⁢

a

n

⁢

(

z

-

z

0

)

n

, with

a

k

≠

0

and

k

>

0

(

a

k

is the first non-zero term).
One can factor the series so that

f

⁢

(

z

)

=

(

z

-

z

0

)

k

⁢

Σ

n

=

0

∞

⁢

a

n

+

k

⁢

(

z

-

z

0

)

n

and define

g

⁢

(

z

)

=

Σ

n

=

0

∞

⁢

a

n

+

k

⁢

(

z

-

z

0

)

n

so that

f

⁢

(

z

)

=

(

z

-

z

0

)

k

⁢

g

⁢

(

z

)

.
Observe that

g

⁢

(

z

)

is analytic on

|

z

-

z

0

|

<

R

.

To show that

z

0

is an isolated zero of

f

,
we must find

ϵ

>

0

so that

f

is non-zero on

0

<

|

z

-

z

0

|

<

ϵ

.
It is enough to find

ϵ

>

0

so that

g

is non-zero
on

|

z

-

z

0

|

<

ϵ

by the

relation

f

⁢

(

z

)

=

(

z

-

z

0

)

k

⁢

g

⁢

(

z

)

.
Because

g

⁢

(

z

)

is analytic, it is

continuous at

z

0

.
Notice that

g

⁢

(

z

0

)

=

a

k

≠

0

,
so there exists an

ϵ

>

0

so that for all

z

with

|

z

-

z

0

|

<

ϵ

it follows that

|

g

⁢

(

z

)

-

a

k

|

<

|

a

k

|

2

.
This implies that

g

⁢

(

z

)

is non-zero in this set.

Title

zeroes of analytic functions are isolated

Canonical name

ZeroesOfAnalyticFunctionsAreIsolated

Date of creation

2013-03-22 13:38:10

Last modified on

2013-03-22 13:38:10

Owner

brianbirgen (2180)

Last modified by

brianbirgen (2180)

Numerical id

8

Author

brianbirgen (2180)

Entry type

Result

Classification

msc 30C15

Synonym

zeros of analytic functions are isolated

Related topic

Complex

Related topic

LeastAndGreatestZero

Related topic

IdentityTheorem

Related topic

WhenAllSingularitiesArePoles

Generated on Fri Feb 9 20:16:32 2018 by

LaTeXML
automorphisms of unit disk

automorphisms of unit disk

All

automorphisms

of the

complex

unit disk

Δ

=

{

z

∈

ℂ

:

|

z

|

<

1

}

to itself,
can be written in the form

f

a

⁢

(

z

)

=

e

i

⁢

θ

⁢

z

-

a

1

-

a

¯

⁢

z

where

a

∈

Δ

and

θ

∈

S

1

.

This map sends

a

to

0

,

1

/

a

¯

to

∞

and the

unit circle

to the unit circle.

Title

automorphisms of unit disk

Canonical name

AutomorphismsOfUnitDisk

Date of creation

2013-03-22 13:36:48

Last modified on

2013-03-22 13:36:48

Owner

brianbirgen (2180)

Last modified by

brianbirgen (2180)

Numerical id

6

Author

brianbirgen (2180)

Entry type

Example

Classification

msc 30C20

Related topic

MobiusTransformation

Related topic

ProofOfConformalMobiusCircleMapTheorem

Generated on Fri Feb 9 20:16:34 2018 by

LaTeXML
unit disk upper half plane conformal equivalence theorem

unit disk upper half plane conformal equivalence theorem

Theorem 1

.

There is a

conformal map

from

Δ

, the

unit disk

, to

U

⁢

H

⁢

P

, the

upper half plane

.

Proof.

Define

f

:

ℂ

^

→

ℂ

^

(where

ℂ

^

denotes the Riemann Sphere) to be

f

⁢

(

z

)

=

z

-

i

z

+

i

. Notice that

f

-

1

⁢

(

w

)

=

i

⁢

1

+

w

1

-

w

and that

f

(and therefore

f

-

1

) is a

Mobius transformation

.

Notice that

f

⁢

(

0

)

=

-

1

,

f

⁢

(

1

)

=

1

-

i

1

+

i

=

-

i

and

f

⁢

(

-

1

)

=

-

1

-

i

-

1

+

i

=

i

. By the

Mobius Circle Transformation Theorem

,

f

takes the

real axis

to the

unit circle

. Since

f

⁢

(

i

)

=

0

,

f

maps

U

⁢

H

⁢

P

to

Δ

and

f

-

1

:

Δ

→

U

⁢

H

⁢

P

. ∎

Title

unit disk upper half plane conformal equivalence theorem

Canonical name

UnitDiskUpperHalfPlaneConformalEquivalenceTheorem

Date of creation

2013-03-22 13:37:52

Last modified on

2013-03-22 13:37:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 30C20

Related topic

UnitDisk

Related topic

UpperHalfPlane

Related topic

MobiusTransformation

Related topic

MobiusCircleTransformationTheorem

Related topic

ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel

Generated on Fri Feb 9 20:16:36 2018 by

LaTeXML
conformal mapping theorem

conformal mapping theorem

Let

f

⁢

(

z

)

be

analytic

in a domain

D

. Then it is

conformal

at any point

z

∈

D

where

f

′

⁢

(

z

)

≠

0

.

Title

conformal mapping theorem

Canonical name

ConformalMappingTheorem

Date of creation

2013-03-22 13:35:44

Last modified on

2013-03-22 13:35:44

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

7

Author

Johan (1032)

Entry type

Theorem

Classification

msc 30C35

Related topic

ConformalMapping

Generated on Fri Feb 9 20:16:37 2018 by

LaTeXML
proof of conformal mapping theorem

proof of conformal mapping theorem

Let

D

⊂

ℂ

be a domain, and let

f

:

D

→

ℂ

be an

analytic function

. By identifying the

complex plane

ℂ

with

ℝ

2

, we can view

f

as a

function

from

ℝ

2

to
itself:

f

~

⁢

(

x

,

y

)

:=

(

ℜ

⁡

f

⁢

(

x

+

i

⁢

y

)

,

ℑ

⁡

f

⁢

(

x

+

i

⁢

y

)

)

=

(

u

⁢

(

x

,

y

)

,

v

⁢

(

x

,

y

)

)

with

u

and

v

real functions. The

Jacobian matrix

of

f

~

is

J

⁢

(

x

,

y

)

=

∂

⁡

(

u

,

v

)

∂

⁡

(

x

,

y

)

=

(

u

x

u

y

v

x

v

y

)

.

As an analytic function,

f

satisfies the

Cauchy-Riemann equations

,
so that

u

x

=

v

y

and

u

y

=

-

v

x

. At a fixed point

z

=

x

+

i

⁢

y

∈

D

, we
can therefore define

a

=

u

x

⁢

(

x

,

y

)

=

v

y

⁢

(

x

,

y

)

and

b

=

u

y

⁢

(

x

,

y

)

=

-

v

x

⁢

(

x

,

y

)

.
We write

(

a

,

b

)

in polar coordinates as

(

r

⁢

cos

⁡

θ

,

r

⁢

sin

⁡

θ

)

and get

J

⁢

(

x

,

y

)

=

(

a

b

-

b

a

)

=

r

⁢

(

cos

⁡

θ

sin

⁡

θ

-

sin

⁡

θ

cos

⁡

θ

)

.

Now we consider two smooth curves through

(

x

,

y

)

, which we
parametrize by

γ

1

⁢

(

t

)

=

(

u

1

⁢

(

t

)

,

v

1

⁢

(

t

)

)

and

γ

2

⁢

(

t

)

=

(

u

2

⁢

(

t

)

,

v

2

⁢

(

t

)

)

. We can choose the parametrization such
that

γ

1

⁢

(

0

)

=

γ

2

⁢

(

0

)

=

z

. The images of these curves under

f

~

are

f

~

∘

γ

1

and

f

~

∘

γ

2

,
respectively, and their

derivatives

at

t

=

0

are

(

f

~

∘

γ

1

)

′

⁢

(

0

)

=

∂

⁡

(

u

,

v

)

∂

⁡

(

x

,

y

)

⁢

(

γ

1

⁢

(

0

)

)

⋅

d

⁢

γ

1

d

⁢

t

⁢

(

0

)

=

J

⁢

(

x

,

y

)

⁢

(

d

⁢

u

1

d

⁢

t

d

⁢

v

1

d

⁢

t

)

and, similarly,

(

f

~

∘

γ

2

)

′

⁢

(

0

)

=

J

⁢

(

x

,

y

)

⁢

(

d

⁢

u

2

d

⁢

t

d

⁢

v

2

d

⁢

t

)

by the

chain rule

. We see that if

f

′

⁢

(

z

)

≠

0

,

f

transforms the

tangent vectors

to

γ

1

and

γ

2

at

t

=

0

(and therefore
in

z

) by the

orthogonal matrix

J

/

r

=

(

cos

⁡

θ

sin

⁡

θ

-

sin

⁡

θ

cos

⁡

θ

)

and scales them by a factor of

r

. In particular, the transformation
by an orthogonal matrix implies that the angle between the tangent
vectors is preserved. Since the

determinant

of

J

/

r

is 1, the
transformation also preserves

orientation

(the

direction

of the angle
between the tangent vectors). We conclude that

f

is a

conformal
mapping

at each point where its derivative is nonzero.

Title

proof of conformal mapping theorem

Canonical name

ProofOfConformalMappingTheorem

Date of creation

2013-03-22 13:47:26

Last modified on

2013-03-22 13:47:26

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

7

Author

pbruin (1001)

Entry type

Proof

Classification

msc 30C35

Generated on Fri Feb 9 20:16:40 2018 by

LaTeXML
Schwarz reflection principle

Schwarz reflection principle

For a region

G

⊂

ℂ

define

G

*

:=

{

z

:

z

¯

∈

G

}

(where

z

¯

is the

complex conjugate

of

z

). If

G

is a

symmetric region

, that is

G

=

G

*

, then we define

G

+

:=

{

z

∈

G

:

Im

⁡

z

>

0

}

,

G

-

:=

{

z

∈

G

:

Im

⁡

z

<

0

}

and

G

0

:=

{

z

∈

G

:

Im

⁡

z

=

0

}

.

Theorem

.

Let

G

⊂

C

be a region such that

G

=

G

*

and suppose that

f

:

G

+

∪

G

0

→

C

is a

continuous functions

that is

analytic

on

G

+

and further that

f

⁢

(

x

)

is real for

x

∈

G

0

(that is
for real

x

), then there is an

analytic function

g

:

G

→

C

such that

g

⁢

(

z

)

=

f

⁢

(

z

)

for

z

∈

G

+

∪

G

0

.

That is you can “reflect” an analytic function across the

real axis

. Note that by composing with various

conformal mappings

you could generalize the above to reflection across an

analytic curve

.
So loosely stated, the

theorem

says that if an analytic function is defined in a region with some “nice”

boundary

and the

function

behaves “nice” on this boundary, then we can extend the function to a larger domain. Let us make this statement precise with the following

generalization

.

Theorem

.

Let

G

,

Ω

⊂

C

be regions and let

γ

and

ω

be

free analytic boundary arcs

in

∂

⁡

G

and

∂

⁡

Ω

. Suppose
that

f

:

G

∪

γ

→

C

is a continuous function that
is analytic on

G

,

f

⁢

(

G

)

⊂

Ω

and

f

⁢

(

γ

)

⊂

ω

, then for any

compact set

κ

⊂

γ

,

f

has an

analytic continuation

to an

open set

containing

G

∪

κ

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

2

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

Schwarz reflection principle

Canonical name

SchwarzReflectionPrinciple

Date of creation

2013-03-22 14:17:58

Last modified on

2013-03-22 14:17:58

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C35

Synonym

Schwarz reflection theorem

Synonym

reflection principle

Defines

symmetric region

Generated on Fri Feb 9 20:16:42 2018 by

LaTeXML
Blaschke product

Blaschke product

Definition

.

Suppose that

{

a

n

}

is a

sequence

of

complex numbers

with

0

<

|

a

n

|

<

1

and

∑

n

=

1

∞

(

1

-

|

a

n

|

)

<

∞

, then

B

⁢

(

z

)

:=

∏

n

=

1

∞

|

a

n

|

a

n

⁢

(

a

n

-

z

1

-

a

¯

n

⁢

z

)

is called the

Blaschke product

.

This

product

converges uniformly on

compact subsets

of the

unit disc

, and thus

B

is
a holomorhic

function

on the unit disc.
Further it is the function on the disc that has zeros exactly at

{

a

n

}

.
And finally for

z

in the unit
disc,

|

B

⁢

(

z

)

|

≤

1

.

Definition

.

Sometimes

B

a

⁢

(

z

)

:=

z

-

a

1

-

a

¯

⁢

z

is called the

Blaschke factor

.

With this definition, the Blascke product becomes

B

⁢

(

z

)

=

∏

n

=

1

∞

|

a

n

|

a

n

⁢

B

a

n

⁢

(

z

)

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Blaschke product

Canonical name

BlaschkeProduct

Date of creation

2013-03-22 14:19:35

Last modified on

2013-03-22 14:19:35

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C45

Defines

Blascke factor

Generated on Fri Feb 9 20:16:44 2018 by

LaTeXML
Koebe 1/4 theorem

Koebe 1/4 theorem

Theorem

(Koebe).

Suppose

f

is a

schlicht function

(

univalent function

on the

unit disc

such that

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

) and

D

⊂

C

is the unit disc in the

complex plane

, then

f

⁢

(

𝔻

)

⊇

{

w

∣

|

w

|

<

1

/

4

}

.

That is, if a univalent function on the unit disc maps 0 to 0 and has derivative
1 at 0, then the image of the unit disc contains the ball of radius

1

/

4

. So for any

w

∉

f

⁢

(

𝔻

)

we have that

|

w

|

≥

1

/

4

. Furthermore, if we look at the

Koebe function

, we can see that the constant

1

/

4

is sharp and cannot be improved.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

Koebe 1/4 theorem

Canonical name

Koebe14Theorem

Date of creation

2013-03-22 14:23:57

Last modified on

2013-03-22 14:23:57

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C45

Synonym

Köbe 1/4 theorem

Synonym

Koebe one-fourth theorem

Synonym

Köbe one-fourth theorem

Related topic

SchlichtFunctions

Generated on Fri Feb 9 20:16:46 2018 by

LaTeXML
Koebe distortion theorem

Koebe distortion theorem

Theorem

(Koebe).

Suppose

f

is a

schlicht function

(

univalent function

on the

unit disc

such that

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

) then

1

-

|

z

|

(

1

+

|

z

|

)

3

≤

|

f

′

⁢

(

z

)

|

≤

1

+

|

z

|

(

1

-

|

z

|

)

3

,

and

|

z

|

(

1

+

|

z

|

)

2

≤

|

f

⁢

(

z

)

|

≤

|

z

|

(

1

-

|

z

|

)

2

.

Equality holds for one of the four

inequalities

at some point

z

≠

0

if and only if

f

is a rotation of the

Koebe function

.

Following is a

generalized distortion theorem

.

Theorem

.

If

K

is a

compact subset

of a region

G

⊂

C

, then there is a constant

M

(depending on

K

) such that for every univalent function

f

on

G

and ever pair of points

z

,

w

∈

K

we have

1

M

≤

|

f

′

⁢

(

z

)

|

|

f

′

⁢

(

w

)

|

≤

M

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

Koebe distortion theorem

Canonical name

KoebeDistortionTheorem

Date of creation

2013-03-22 14:23:25

Last modified on

2013-03-22 14:23:25

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C45

Synonym

distortion theorem

Synonym

generalized distortion theorem

Synonym

Köbe distortion theorem

Related topic

SchlichtFunctions

Generated on Fri Feb 9 20:16:48 2018 by

LaTeXML