aarjaneiro/planetmath huggingface dataset:
Koebe function

Koebe function

Definition

.

The

analytic function

f

⁢

(

z

)

:=

z

(

1

-

z

)

2

on the

unit disc

in the

complex plane

is called the

Koebe function

. For some

|

α

|

=

1

, the

functions

f

α

⁢

(

z

)

:=

z

(

1

-

α

⁢

z

)

2

are called

rotations of the Koebe function

.

Firstly note that

f

1

=

f

, and
next
note that

f

is a map from the open unit disc onto

ℂ

\

(

-

∞

,

-

1

/

4

]

. The maps

f

α

⁢

(

z

)

can be also given as

f

α

⁢

(

z

)

=

α

¯

⁢

f

1

⁢

(

α

⁢

z

)

.
Further note that the

power series

representation

of these
functions is given by

f

α

⁢

(

z

)

=

z

(

1

-

α

⁢

z

)

2

=

∑

n

=

1

∞

n

⁢

α

n

-

1

⁢

z

n

.

Also note that these functions belong to the class of

Schlicht functions

.

References

1

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

Title

Koebe function

Canonical name

KoebeFunction

Date of creation

2013-03-22 14:23:30

Last modified on

2013-03-22 14:23:30

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C45

Synonym

Köbe function

Defines

rotation of the Koebe function

Defines

rotation of the Köbe function

Generated on Fri Feb 9 20:16:50 2018 by

LaTeXML
schlicht functions

schlicht functions

Definition

.

The class of

univalent functions

on the open

unit disc

in the

complex plane

such that for any

f

in the class we have

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

is called
the class of

schlicht functions

. Usually this class is denoted by

𝒮

.

Note that if

g

is any univalent function on the unit disc, then the

function

f

defined by

f

⁢

(

z

)

:=

g

⁢

(

z

)

-

g

⁢

(

0

)

g

′

⁢

(

0

)

belongs to

𝒮

. So to study univalent functions on the unit disc it suffices to study

𝒮

. A basic result on these gives that this set is in fact

compact

in the

space of analytic functions

on the unit disc.

Theorem

.

Let

{

f

n

}

be a sequence of functions in

S

and

f

n

→

f

uniformly on compact subsets of the open unit disc. Then

f

is in

S

.

Alternatively this

theorem

can be stated for all univalent functions by the above remark, but there a sequence of univalent functions can converge either
to a univalent function or to a constant. The requirement that the

first derivative

is 1 for functions in

𝒮

prevents this problem.

References

1

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

Title

schlicht functions

Canonical

name

SchlichtFunctions

Date of creation

2013-03-22 14:23:37

Last modified on

2013-03-22 14:23:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C45

Synonym

schlicht function

Related topic

KoebeDistortionTheorem

Related topic

Koebe14Theorem

Generated on Fri Feb 9 20:16:52 2018 by

LaTeXML
Bieberbach’s conjecture

Bieberbach’s conjecture

The following theorem is known as the

Bieberbach conjecture

, even though it has
now been proven. Bieberbach proposed it in 1916 and it was finally proven in 1984 by Louis de Branges.

Firstly note that if

f

:

𝔻

→

ℂ

is a

schlicht function

(

univalent

,

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

) then

f

has a

power series

representation

as

f

⁢

(

z

)

=

z

+

a

2

⁢

z

2

+

a

3

⁢

z

3

+

⋯

=

z

+

∑

k

=

2

∞

a

k

⁢

z

k

.

Theorem

(Bieberbach).

Suppose that

f

is a schlicht function, then

|

a

k

|

≤

k

for
all

k

≥

2

and furthermore
if there is some integer

k

such that

|

a

k

|

=

k

, then

f

is some rotation of the

Koebe function

.

In fact if

f

is a rotation of the Koebe function then

|

a

k

|

=

k

for all

k

.

References

1

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

Title

Bieberbach’s conjecture

Canonical name

BieberbachsConjecture

Date of creation

2013-03-22 14:24:07

Last modified on

2013-03-22 14:24:07

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C55

Classification

msc 30C45

Synonym

Bieberbach conjecture

Generated on Fri Feb 9 20:16:54 2018 by

LaTeXML
Loewner chain

Loewner chain

A

Loewner chain

is a

continuous function

f

:

𝔻

×

[

0

,

∞

)

→

ℂ

(where

𝔻

refers to the open

unit disc

) with the following properties:

1.

the

function

z

↦

f

⁢

(

z

,

t

)

is

analytic

and

univalent

for all

t

∈

[

0

,

∞

)

2.

f

⁢

(

0

,

t

)

=

0

and

f

′

⁢

(

0

,

t

)

=

e

t

3.

f

⁢

(

𝔻

,

s

)

⊆

f

⁢

(

𝔻

,

t

)

for

0

≤

s

<

t

<

∞

Bibliography

•

John B. Conway

. Springer-Verlag, New York, Inc, 1995

Title

Loewner chain

Canonical name

LoewnerChain

Date of creation

2013-03-22 14:12:02

Last modified on

2013-03-22 14:12:02

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C55

Generated on Fri Feb 9 20:16:58 2018 by

LaTeXML
univalent analytic function

univalent analytic function

Definition

.

An

analytic function

on an open set is called

univalent

if it is

one-to-one

.

For example

mappings

of the

unit disc

to itself

ϕ

a

:

𝔻

→

𝔻

, where

ϕ

a

⁢

(

z

)

=

z

-

a

1

-

a

¯

⁢

z

, for any

a

∈

𝔻

are univalent.
The following summarizes some
basic of univalent functions.

Proposition

.

Suppose that

G

,

Ω

⊂

C

are regions and

f

:

G

→

Ω

is a univalent mapping such that

f

⁢

(

G

)

=

Ω

(it
is onto), then

•

f

-

1

:

Ω

→

G

(where

f

-

1

⁢

(

f

⁢

(

z

)

)

=

z

) is an analytic
function and

(

f

-

1

)

′

⁢

(

f

⁢

(

z

)

)

=

1

f

′

⁢

(

z

)

,

•

f

′

⁢

(

z

)

≠

0

for all

z

∈

G

References

1

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

2

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

Title

univalent analytic function

Canonical name

UnivalentAnalyticFunction

Date of creation

2013-03-22 14:12:06

Last modified on

2013-03-22 14:12:06

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C55

Synonym

univalent function

Synonym

univalent

Generated on Fri Feb 9 20:17:00 2018 by

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Beurling-Ahlfors quasiconformal extension

Beurling-Ahlfors quasiconformal extension

Theorem

(Beurling-Ahlfors).

There exists a

quasiconformal mapping

of the

upper half plane

to itself if and only if the boundary correspondence

mapping

μ

is

M

-

quasisymmetric

(

http://planetmath.org/QuasisymmetricMapping

). Furthermore there exists an

extension

of

μ

to a quasiconformal mapping of the upper half planes such that the maximal dilatation of the extension depends only on

M

and not on

μ

.

That is, the extension is

K

-quasiconformal (

http://planetmath.org/QuasiconformalMapping

) if and only if the boundary correspondence is

M

-quasisymmetric (

http://planetmath.org/QuasisymmetricMapping

) where

K

depends purely on

M

.

Supposing that we have the mapping

ϕ

:

H

→

H

(where

H

is the upper half plane), then the mapping

μ

:

ℝ

→

ℝ

,
such that

μ

⁢

(

x

)

=

ϕ

⁢

(

x

)

where

x

∈

ℝ

, is the boundary correspondence of

ϕ

.

To prove the

sufficiency

of the above

theorem

Beurling and Ahlfors

[

2

]

define

ϕ

as follows. Given a

μ

that is a quasisymmetric mapping of the real line onto itself and fixes

∞

, we define a map

ϕ

⁢

(

x

,

y

)

=

u

⁢

(

x

,

y

)

+

i

⁢

u

⁢

(

x

,

y

)

where

u

⁢

(

x

,

y

)

=

1

2

⁢

y

⁢

∫

-

y

y

μ

⁢

(

x

+

t

)

⁢

𝑑

t

,

v

⁢

(

x

,

y

)

=

1

2

⁢

y

⁢

∫

0

y

(

μ

⁢

(

x

+

t

)

-

μ

⁢

(

x

-

t

)

)

⁢

𝑑

t

.

Intuitively

ϕ

is a

function

which “smoothes” out any kinks in the function

μ

as we get further and further away from the real line. It therefore intuitively follows that

ϕ

has the worst (highest) dilatation near the

x

axis, which actually turns out to be true.

References

1

L. V. Ahlfors.

. Van Nostrand-Reinhold, Princeton, New Jersey, 1966

2

A. Beurling, L. V. Ahlfors. .

Acta Math.

, 96:125-142,
1956.

3

J. Lebl.

. . Also available at

http://www.jirka.org/thesis.pdf

http://www.jirka.org/thesis.pdf

Title

Beurling-Ahlfors quasiconformal extension

Canonical name

BeurlingAhlforsQuasiconformalExtension

Date of creation

2013-03-22 14:06:49

Last modified on

2013-03-22 14:06:49

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

12

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C62

Synonym

Beurling-Ahlfors theorem

Related topic

QuasiconformalMapping

Related topic

QuasisymmetricMapping

Generated on Fri Feb 9 20:17:02 2018 by

LaTeXML
quasicircle

quasicircle

If

f

:

ℂ

→

ℂ

is a

quasiconformal mapping

with maximal dilatation of

K

,
then

f

⁢

(

S

1

)

is called a

quasicircle

or

K

-quasicircle.

If

f

:

ℝ

n

→

ℝ

n

, for

n

>

2

is a quasiconformal mapping with maximal dilatation

K

, then we call

f

⁢

(

S

n

-

1

)

a

quasisphere

or

K

-quasisphere.

An example of a quasicircle is the famous

Koch snowflake

.

Title

quasicircle

Canonical name

Quasicircle

Date of creation

2013-03-22 14:10:41

Last modified on

2013-03-22 14:10:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C62

Classification

msc 30C65

Synonym

quasisphere

Synonym

K-quasicircle

Synonym

K-quasisphere

Defines

quasicircle

Defines

quasisphere

Defines

K-quasicircle

Defines

K-quasisphere

Generated on Fri Feb 9 20:17:04 2018 by

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quasisymmetric mapping

quasisymmetric mapping

A

function

μ

of the real line to itself is

quasisymmetric

(or

M

-quasisymmetric

) if it

satisfies

the following

M

-condition.

There exists an

M

, such that for all

x

,

t

(where

t

≠

x

)

1

M

≤

μ

⁢

(

x

+

t

)

-

μ

⁢

(

x

)

μ

⁢

(

x

)

-

μ

⁢

(

x

-

t

)

≤

M

.

Geometrically this means that the ratio of the length of the

intervals

μ

⁢

[

(

x

-

t

,

x

)

]

and

μ

⁢

[

(

x

,

x

+

t

)

]

is

bounded

. This implies among other things that the function is

one-to-one

and

continuous

.

For example powers (as long as you make them one-to-one by for example using an odd power, or defining them as

-

|

x

|

p

for

negative

x

and

|

x

|

p

for positive

x

where

p

>

0

) are quasisymmetric. On the other hand functions like

e

x

-

e

-

x

, while one-to-one, onto and continuous, are not quasisymmetric. It would seem like a very strict condition, however it has been shown that there in fact exist functions that are quasisymmetric, but are not even

absolutely continuous

.

Quasisymmetric functions are an analogue of

quasiconformal mappings

.

Title

quasisymmetric mapping

Canonical

name

QuasisymmetricMapping

Date of creation

2013-03-22 14:06:45

Last modified on

2013-03-22 14:06:45

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C65

Classification

msc 26A15

Classification

msc 26A12

Related topic

QuasiconformalMapping

Related topic

BeurlingAhlforsQuasiconformalExtension

Defines

M

-condition

Defines

quasisymmetric

Defines

M

-quasisymmetric

Generated on Fri Feb 9 20:17:08 2018 by

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Hardy’s theorem

Hardy’s theorem

Theorem

.

Let

f

be a

holomorphic function

on

B

⁢

(

0

,

R

)

(the open ball of radius

R

)
and

f

is not a constant function, then

I

⁢

(

r

)

:=

1

2

⁢

π

⁢

∫

0

2

⁢

π

|

f

⁢

(

r

⁢

e

i

⁢

θ

)

|

⁢

𝑑

θ

is

strictly increasing

and

log

⁡

I

⁢

(

r

)

is a

convex function

of

log

⁡

r

.

References

1

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

Title

Hardy’s theorem

Canonical name

HardysTheorem

Date of creation

2013-03-22 14:19:44

Last modified on

2013-03-22 14:19:44

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C80

Classification

msc 30E20

Related topic

HadamardThreeCircleTheorem

Generated on Fri Feb 9 20:17:10 2018 by

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maximum principle

maximum principle

Maximum principle

Let

f

:

U

→

ℝ

(where

U

⊆

ℝ

d

) be a

harmonic function

. Then

f

attains its extremal values on any

compact

K

⊆

U

on the boundary

∂

⁡

K

of

K

. If

f

attains an extremal value anywhere in the

interior

of

K

, then it is constant.

Maximal modulus principle

Let

f

:

U

→

ℂ

(where

U

⊆

ℂ

) be a

holomorphic function

. Then

|

f

|

attains its maximal value on any compact

K

⊆

U

on the boundary

∂

⁡

K

of

K

. If

|

f

|

attains its maximal value anywhere on the

interior

of

K

, then it is constant.

Title

maximum principle

Canonical name

MaximumPrinciple

Date of creation

2013-03-22 12:46:06

Last modified on

2013-03-22 12:46:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 30C80

Classification

msc 31A05

Classification

msc 31B05

Classification

msc 30F15

Synonym

maximal modulus principle

Synonym

maximum principle for harmonic functions

Related topic

HadamardThreeCircleTheorem

Related topic

PhragmenLindelofTheorem

Generated on Fri Feb 9 20:17:11 2018 by

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Phragmén-Lindelöf theorem

Phragmén-Lindelöf theorem

First some notation. Let

∂

∞

⁡

G

be the

extended boundary

of

G

. That is, the

boundary

of

G

, plus optionally the

point at infinity

if in fact

G

is

unbounded

.

Theorem

.

Let

G

be a simply connected region and let

f

:

G

→

C

and

φ

:

G

→

C

be

analytic functions

. Furthermore
suppose that

φ

never vanishes and is

bounded

on

G

. If

M

is a

constant

and

∂

∞

⁡

G

=

A

∪

B

such that

1.

for every

a

∈

A

,

lim

¯

z

→

a

⁡

|

f

⁢

(

z

)

|

≤

M

, and

2.

for every

b

∈

B

, and

η

>

0

,

lim

¯

z

→

b

⁡

|

f

⁢

(

z

)

|

⁢

|

φ

⁢

(

z

)

|

η

≤

M

,

then

|

f

⁢

(

z

)

|

≤

M

for all

z

∈

G

.

This

theorem

is a

generalization

of the

maximal modulus principle

, but instead of requiring that the

function

is bounded as we approach the boundary, we only need a

restriction

on its growth to it to in fact be bounded in all of

G

.

If you let

A

=

∂

∞

⁡

G

(and

φ

≡

1

perhaps), then you get almost exactly one version of the maximal modulus principle. In this case it turns out that

G

need not be simply connected since that is only needed to
define

z

↦

φ

⁢

(

z

)

η

.

In fact the requirement that

G

be simply connected can be eased up a bit in this theorem since it is only needed locally. So the theorem is still true if for every point

z

∈

∂

∞

⁡

G

there exists an open

neighbourhood

N

of

z

such
that

N

∩

G

is simply connected.

References

1

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

Title

Phragmén-Lindelöf theorem

Canonical name

PhragmenLindelofTheorem

Date of creation

2013-03-22 14:12:09

Last modified on

2013-03-22 14:12:09

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C80

Synonym

Phragmén-Lindelöf principle

Related topic

MaximalModulusPrinciple

Generated on Fri Feb 9 20:17:13 2018 by

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proof of Hadamard three-circle theorem

proof of Hadamard three-circle theorem

Let

f

be

holomorphic

on a

closed annulus

0

<

r

1

≤

|

z

|

≤

r

2

. Let

s

=

log

⁡

r

1

-

log

⁡

r

log

⁡

r

2

-

log

⁡

r

1

.

Let

M

⁢

(

r

)

=

M

f

⁢

(

r

)

=

||

f

||

r

=

max

|

z

|

=

r

⁡

|

f

⁢

(

z

)

|

. Then we have to prove that

log

⁡

M

⁢

(

r

)

≤

(

1

-

s

)

⁢

log

⁡

M

⁢

(

r

1

)

+

s

⁢

log

⁡

M

⁢

(

r

2

)

.

For this, let

α

be a

real number

; the

function

α

⁢

log

⁡

|

z

|

+

log

⁡

|

f

⁢

(

z

)

|

is

harmonic

outside the zeros of

f

. Near the zeros of

f

the above function has values which are large negative. Hence by the

maximum modulus principle

this function has its

maximum

on the

boundary

of the annulus, specifically on the two circles

|

z

|

=

r

1

and

|

z

|

=

r

2

. Therefore

α

⁢

log

⁡

|

z

|

+

log

⁡

|

f

⁢

(

z

)

|

≤

max

⁡

(

α

⁢

log

⁡

r

1

+

log

⁡

M

⁢

(

r

1

)

,

α

⁢

log

⁡

r

2

+

log

⁡

M

⁢

(

r

2

)

)

for all

z

in the annulus. In particular, we get the

inequality

α

⁢

log

⁡

r

+

log

⁡

M

⁢

(

r

)

≤

max

⁡

(

α

⁢

log

⁡

r

1

+

log

⁡

M

⁢

(

r

1

)

,

α

⁢

log

⁡

r

2

+

log

⁡

M

⁢

(

r

2

)

)

.

Now let

α

be such that the two values inside the parentheses on the right are equal, that is

α

=

log

⁡

M

⁢

(

r

2

)

-

log

⁡

M

⁢

(

r

1

)

log

⁡

r

1

-

log

⁡

r

2

.

Then from the previous inequality, we get

log

⁡

M

⁢

(

r

)

≤

α

⁢

log

⁡

r

1

+

log

⁡

M

⁢

(

r

1

)

-

α

⁢

log

⁡

r

,

which upon substituting the value for

α

gives the result stated in the theorem.

References

Lang, S.

Complex analysis

, Fourth edition

. Graduate Texts in Mathematics, 103.
Springer-Verlag, New York, 1999. xiv+485 pp. ISBN 0-387-98592-1

Title

proof of Hadamard three-circle theorem

Canonical name

ProofOfHadamardThreecircleTheorem

Date of creation

2013-03-22 15:56:02

Last modified on

2013-03-22 15:56:02

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

5

Author

Simone (5904)

Entry type

Proof

Classification

msc 30C80

Classification

msc 30A10

Generated on Fri Feb 9 20:17:15 2018 by

LaTeXML
proof of Schwarz lemma

proof of Schwarz lemma

Define

g

⁢

(

z

)

=

f

⁢

(

z

)

/

z

. Then

g

:

Δ

→

ℂ

is a

holomorphic function

. The

Schwarz lemma

is just an application of the

maximal modulus principle

to

g

.

For any

1

>

ϵ

>

0

, by the maximal modulus principle

|

g

|

must attain its maximum on the closed disk

{

z

:

|

z

|

≤

1

-

ϵ

}

at its boundary

{

z

:

|

z

|

=

1

-

ϵ

}

, say at some point

z

ϵ

. But then

|

g

⁢

(

z

)

|

≤

|

g

⁢

(

z

ϵ

)

|

≤

1

1

-

ϵ

for any

|

z

|

≤

1

-

ϵ

. Taking an infinimum as

ϵ

→

0

, we see that values of

g

are

bounded

:

|

g

⁢

(

z

)

|

≤

1

.

Thus

|

f

⁢

(

z

)

|

≤

|

z

|

. Additionally,

f

′

⁢

(

0

)

=

g

⁢

(

0

)

, so we see that

|

f

′

⁢

(

0

)

|

=

|

g

⁢

(

0

)

|

≤

1

. This is the first part of the lemma.

Now suppose, as per the premise of the second part of the lemma, that

|

g

⁢

(

w

)

|

=

1

for some

w

∈

Δ

. For any

r

>

|

w

|

, it must be that

|

g

|

attains its maximal

modulus

(1)

inside

the disk

{

z

:

|

z

|

≤

r

}

, and it follows that

g

must be constant inside the entire open disk

Δ

. So

g

⁢

(

z

)

≡

a

for

a

=

g

⁢

(

w

)

of modulus 1, and

f

⁢

(

z

)

=

a

⁢

z

, as required.

Title

proof of Schwarz lemma

Canonical name

ProofOfSchwarzLemma

Date of creation

2013-03-22 12:45:07

Last modified on

2013-03-22 12:45:07

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 30C80

Generated on Fri Feb 9 20:17:17 2018 by

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Schwarz lemma

Schwarz lemma

Let

Δ

=

{

z

:

|

z

|

<

1

}

be the open unit disk in the

complex plane

ℂ

. Let

f

:

Δ

→

Δ

be a

holomorphic function

with

f

⁢

(

0

)

=

0

.
Then

|

f

⁢

(

z

)

|

≤

|

z

|

for all

z

∈

Δ

, and

|

f

′

⁢

(

0

)

|

≤

1

. If the equality

|

f

⁢

(

z

)

|

=

|

z

|

holds for

any

z

≠

0

or

|

f

′

⁢

(

0

)

|

=

1

, then

f

is a rotation:

f

⁢

(

z

)

=

a

⁢

z

with

|

a

|

=

1

.

This lemma is less celebrated than the bigger guns (such as the

Riemann mapping theorem

, which it helps prove); however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No result exists for real functions, of course.

Title

Schwarz lemma

Canonical name

SchwarzLemma

Date of creation

2013-03-22 12:44:37

Last modified on

2013-03-22 12:44:37

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 30C80

Generated on Fri Feb 9 20:17:19 2018 by

LaTeXML
Montel’s theorem

Montel’s theorem

Suppose that

G

⊂

ℂ

is a region.

Theorem

(Montel).

A set

F

of

holomorphic functions

f

:

G

→

C

is normal (

http://planetmath.org/NormalFamily

) if and only if

F

is

locally bounded

.

In other words a sequence of holomorphic functions

{

f

n

}

has a

subsequence

which converges uniformly
on

compact subsets

to a holomorphic function

f

:

G

→

ℂ

if and only if the set

{

f

n

}

is locally bounded.

References

1

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

Title

Montel’s

theorem

Canonical

name

MontelsTheorem

Date of creation

2013-03-22 14:17:52

Last modified on

2013-03-22 14:17:52

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C99

Related topic

AscoliArzelaTheorem

Related topic

SpaceOfAnalyticFunctions

Generated on Fri Feb 9 20:17:21 2018 by

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addition formula

addition formula

The

addition formula

of a real (

http://planetmath.org/RealFunction

) or

complex function

shows how the value of the

function

on a sum-formed variable can be expressed with the values of this function and perhaps of another function on the addends.

Examples

1.

Addition formula of an

additive function

f

,

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

2.

Addition formula of the

natural power function

, i.e. the

binomial theorem

,

(

x

+

y

)

n

=

∑

j

=

0

n

(

n

j

)

x

n

-

j

y

j

(

n

=

0

,

1

,

2

,

…

)

3.

Addition formula of the

exponential function

(

http://planetmath.org/ComplexExponentialFunction

),

e

x

+

y

=

e

x

⁢

e

y

4.

Addition formulae

of the

trigonometric functions

(

http://planetmath.org/DefinitionsInTrigonometry

), e.g.

cos

⁡

(

x

+

y

)

=

cos

⁡

x

⁢

cos

⁡

y

-

sin

⁡

x

⁢

sin

⁡

y

,

1

1

The addition formula of cosine is sometimes called “the mother of all formulae”.

⁢

tan

⁡

(

x

+

y

)

=

tan

⁡

x

+

tan

⁡

y

1

-

tan

⁡

x

⁢

tan

⁡

y

5.

Addition formulae of the

hyperbolic functions

, e.g.

sinh

⁡

(

x

+

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

+

cosh

⁡

x

⁢

sinh

⁡

y

6.

Addition formula of the

Bessel function

,

J

n

(

x

+

y

)

=

∑

ν

=

-

∞

∞

J

ν

(

x

)

J

n

-

ν

(

y

)

(

n

=

0

,

±

1

,

±

2

,

…

)

The five first of those are instances of ; e.g.

cosh

⁡

x

and

sinh

⁡

x

are tied together by the

algebraic

connection (

http://planetmath.org/UnitHyperbola

)

cosh

2

⁡

x

-

sinh

2

⁡

x

=

1

.

One may also speak of the

subtraction formulae

of functions — one example would be

e

x

-

y

=

e

x

e

y

.

Title

addition formula

Canonical name

AdditionFormula

Date of creation

2013-03-22 14:44:20

Last modified on

2013-03-22 14:44:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D05

Classification

msc 30A99

Classification

msc 26A99

Synonym

addition theorem

Defines

addition formulae

Defines

subtraction formula

Defines

subtraction formulae

Generated on Fri Feb 9 20:17:23 2018 by

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