aarjaneiro/planetmath huggingface dataset:
proof of Marty’s theorem

proof of Marty’s theorem

(i)

Fix

K

⊆

Ω

compact

. We have:

2

⁢

|

f

′

⁢

(

z

)

|

1

+

|

f

⁢

(

z

)

|

2

≤

M

K

⁢

∀

f

∈

ℱ

,

z

∈

K

($*$)

Let

V

be a region with

K

=

V

¯

and let

γ

:

[

a

,

b

]

→

V

be the

C

1

curves connecting the points

P

,

Q

∈

Ω

. Then we have:

d

σ

⁢

(

f

⁢

(

P

)

,

f

⁢

(

Q

)

)

=

inf

γ

⁡

l

σ

⁢

(

f

∘

γ

)

=

inf

γ

⁡

∫

a

b

∥

(

f

∘

γ

)

′

⁢

(

t

)

∥

σ

,

f

∘

γ

⁢

(

t

)

⁢

𝑑

t

=

inf

γ

⁡

∫

a

b

2

⁢

|

f

′

⁢

(

γ

⁢

(

t

)

)

|

1

+

|

f

⁢

(

γ

⁢

(

t

)

)

|

2

⁢

|

γ

′

⁢

(

t

)

|

⁢

𝑑

t

(

⁢

($*$)

‣

proof of Marty’s theorem

⁢

)

≤

M

K

⁢

inf

γ

⁡

∫

a

b

|

γ

′

⁢

(

t

)

|

⁢

𝑑

t

=

M

K

⁢

inf

γ

⁡

l

⁢

(

γ

)

=

M

K

⁢

|

P

-

Q

|

Thus

f

is

Lipschitz continuous

and thus

ℱ

is

equicontinuous

. By the Ascoli-ArzelÃÂ

Theorem

we conclude that

ℱ

is normal.

(ii)

Now assume

ℱ

to be normal. Define:

f

♯

⁢

(

z

)

:=

2

⁢

|

f

′

⁢

(

z

)

|

1

+

|

f

⁢

(

z

)

|

2

Let

K

⊆

Ω

be compact. To obtain

contradiction

assume

{

f

♯

:

f

∈

ℱ

}

is not uniformly bounded on

K

.
But then there exists a

sequence

{

f

n

}

⊂

ℱ

such that:

max

z

∈

K

⁡

f

n

♯

⁢

(

z

)

→

∞

(

n

→

∞

)

Since

ℱ

is normal for each

P

∈

Ω

let there be a

neighbourhood

U

P

such that

{

f

n

}

converges

normally to a meromorphic function

f

. But from

(

1

/

f

n

)

♯

=

f

n

♯

we see that

{

f

n

♯

}

converges normally on

U

P

. Since

K

is compact it can be covered by finitely many sets

U

P

. We conclude that

{

f

n

♯

}

must be

bounded

on

K

and obtain a contradiction. ∎

Title

proof of Marty’s theorem

Canonical name

ProofOfMartysTheorem

Date of creation

2013-03-22 18:23:11

Last modified on

2013-03-22 18:23:11

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

4

Author

karstenb (16623)

Entry type

Proof

Classification

msc 30D30

Generated on Fri Feb 9 20:19:05 2018 by

LaTeXML
proof of Riemann’s removable singularity theorem

proof of Riemann’s removable singularity theorem

Suppose that

f

is

holomorphic

on

U

∖

{

a

}

and

lim

z

→

a

⁡

(

z

-

a

)

⁢

f

⁢

(

z

)

=

0

. Let

f

⁢

(

z

)

=

∑

k

=

-

∞

∞

c

k

⁢

(

z

-

a

)

k

be the

Laurent series

of

f

centered at

a

. We will show that

c

k

=

0

for

k

<

0

, so that

f

can be holomorphically extended to all of

U

by defining

f

⁢

(

a

)

=

c

0

.

For any non-negative integer

n

, the

residue

of

(

z

-

a

)

n

⁢

f

⁢

(

z

)

at

a

is

Res

⁡

(

(

z

-

a

)

n

⁢

f

⁢

(

z

)

,

a

)

=

1

2

⁢

π

⁢

i

⁢

lim

δ

→

0

+

⁡

∮

|

z

-

a

|

=

δ

(

z

-

a

)

n

⁢

f

⁢

(

z

)

⁢

d

z

.

This is equal to zero, because

|

∮

|

z

-

a

|

=

δ

(

z

-

a

)

n

⁢

f

⁢

(

z

)

⁢

d

z

|

≤

2

⁢

π

⁢

δ

⁢

max

|

z

-

a

|

=

δ

⁡

|

(

z

-

a

)

n

⁢

f

⁢

(

z

)

|

=

2

⁢

π

⁢

δ

n

⁢

max

|

z

-

a

|

=

δ

⁡

|

(

z

-

a

)

⁢

f

⁢

(

z

)

|

which, by our

assumption

, goes to zero as

δ

→

0

. Since the residue of

(

z

-

a

)

n

⁢

f

⁢

(

z

)

at

a

is also equal to

c

-

n

-

1

, the coefficients of all

negative

powers of

z

in the Laurent series vanish.

Conversely

, if

a

is a

removable singularity

of

f

, then

f

can be expanded in a

power series

centered at

a

, so that

lim

z

→

a

⁡

(

z

-

a

)

⁢

f

⁢

(

z

)

=

0

because the constant term in the power series of

(

z

-

a

)

⁢

f

⁢

(

z

)

is zero.

A

corollary

of this

theorem

is the following: if

f

is

bounded

near

a

, then

|

(

z

-

a

)

⁢

f

⁢

(

z

)

|

≤

|

z

-

a

|

⁢

M

for some

M

>

0

. This implies that

(

z

-

a

)

⁢

f

⁢

(

z

)

→

0

as

z

→

a

, so

a

is a removable singularity of

f

.

Title

proof of Riemann’s removable singularity theorem

Canonical

name

ProofOfRiemannsRemovableSingularityTheorem

Date of creation

2013-03-22 13:33:03

Last modified on

2013-03-22 13:33:03

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

5

Author

pbruin (1001)

Entry type

Proof

Classification

msc 30D30

Generated on Fri Feb 9 20:19:08 2018 by

LaTeXML
residue

residue

Let

U

⊂

ℂ

be a domain and let

f

:

U

⟶

ℂ

be a

function

represented by a

Laurent series

f

⁢

(

z

)

:=

∑

k

=

-

∞

∞

c

k

⁢

(

z

-

a

)

k

centered about

a

. The coefficient

c

-

1

of the above Laurent series is called the

residue

of

f

at

a

, and denoted

Res

⁡

(

f

;

a

)

.

Title

residue

Canonical

name

Residue

Date of creation

2013-03-22 12:04:56

Last modified on

2013-03-22 12:04:56

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 30D30

Related topic

CauchyResidueTheorem

Generated on Fri Feb 9 20:19:09 2018 by

LaTeXML
residue at infinity

residue at infinity

If in the

Laurent expansion

f

⁢

(

z

)

=

∑

k

=

-

∞

∞

c

k

⁢

z

k

(1)

of the

function

f

, the

coefficient

c

n

is distinct from zero (

n

>

0

) and

c

n

+

1

=

c

n

+

2

=

…

=

0

,  then there exists the numbers

M

and

K

such that

|

z

-

n

⁢

f

⁢

(

z

)

|

<

M

always when

|

z

|

>

K

.

In this case one says that

∞

is a

pole of order

n

of the function

f

(cf.

zeros and poles of rational function

).

If there is no such positive integer

n

, (1) infinitely many positive powers of

z

, and one may say that

∞

is an

essential singularity

of

f

.

In both cases one can define for

f

the

residue at infinity

as

1

2

⁢

i

⁢

π

⁢

∮

C

f

⁢

(

z

)

⁢

𝑑

z

=

c

-

1

,

(2)

where the

integral

is taken along a closed

contour

C

which goes once anticlockwise around the origin, i.e. once clockwise around the point

z

=

∞

(see the

Riemann sphere

).

Then the usual form

1

2

⁢

i

⁢

π

⁢

∮

C

f

⁢

(

z

)

⁢

𝑑

z

=

∑

j

Res

⁢

(

f

;

a

j

)

of the

residue theorem

may be expressed as follows:

The sum of all

residues

of an

analytic function

having only a finite number of points of

singularity

is equal to zero.

References

1

Ernst Lindelöf

:

Le calcul des résidus et ses applications à la théorie des fonctions

.  Gauthier-Villars, Paris (1905).

Title

residue at infinity

Canonical name

ResidueAtInfinity

Date of creation

2013-03-22 19:15:00

Last modified on

2013-03-22 19:15:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D30

Related topic

Residue

Related topic

RegularAtInfinity

Generated on Fri Feb 9 20:19:11 2018 by

LaTeXML
Riemann’s removable singularity theorem

Riemann’s removable singularity theorem

Let

U

⊂

ℂ

be a domain,

a

∈

U

, and let

f

:

U

∖

{

a

}

be

holomorphic

. Then

a

is a

removable singularity

of

f

if and only if

lim

z

→

a

⁡

(

z

-

a

)

⁢

f

⁢

(

z

)

=

0

.

In particular,

a

is a removable singularity of

f

if

f

is

http://planetmath.org/node/

Bounded

bounded near

a

, i.e. if there is a punctured

neighborhood

V

of

a

and a

real number

M

>

0

such that

|

f

⁢

(

z

)

|

<

M

for all

z

∈

V

.

Title

Riemann’s removable singularity theorem

Canonical name

RiemannsRemovableSingularityTheorem

Date of creation

2013-03-22 13:33:00

Last modified on

2013-03-22 13:33:00

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Theorem

Classification

msc 30D30

Related topic

Pole

Related topic

EssentialSingularity

Related topic

Meromorphic

Related topic

RiemannsTheoremOnIsolatedSingularities

Generated on Fri Feb 9 20:19:13 2018 by

LaTeXML
Riemann’s removable singularity theorem in several variables

Riemann’s removable singularity theorem in several variables

Theorem

.

Suppose

V

is a proper

analytic variety

in an

open set

U

⊂

C

n

(that is of

dimension

at most

n

-

1

)
suppose that

f

:

U

\

V

→

C

is

holomorphic

and further that

f

is

locally bounded

in

U

Then there exists a unique holomorphic
extention of

f

to all of

U

.

If

V

is of even lower dimension we can in fact even drop the locally bounded
requirement, see the

Hartogs extension theorem

.

References

1

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

2

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

Riemann’s

removable singularity

theorem in several variables

Canonical name

RiemannsRemovableSingularityTheoremInSeveralVariables

Date of creation

2013-03-22 15:34:57

Last modified on

2013-03-22 15:34:57

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30D30

Classification

msc 32H02

Synonym

Riemann’s extension theorem

Generated on Fri Feb 9 20:19:14 2018 by

LaTeXML
simple pole

simple pole

A

simple pole

is a pole of order 1. That is, a

meromorphic function

f

has a simple pole at

x

0

∈

ℂ

if

f

⁢

(

z

)

=

a

z

-

x

0

+

g

⁢

(

z

)

where

a

≠

0

∈

ℂ

, and

g

is

holomorphic

at

x

0

.

Note that

a

in the equation above is the

residue

of

f

at

x

0

.

Title

simple pole

Canonical

name

SimplePole

Date of creation

2013-03-22 13:15:55

Last modified on

2013-03-22 13:15:55

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 30D30

Generated on Fri Feb 9 20:19:16 2018 by

LaTeXML
spherical derivative

spherical derivative

Let

G

⊂

ℂ

be a domain.

Definition

.

Let

f

:

G

→

ℂ

be a

meromorphic function

, then the

spherical
derivative

of

f

, denoted

f

♯

is defined as

f

♯

⁢

(

z

)

:=

2

⁢

|

f

′

⁢

(

z

)

|

1

+

|

f

⁢

(

z

)

|

2

for

z

where

f

⁢

(

z

)

≠

∞

and when

f

⁢

(

z

)

=

∞

define

f

♯

⁢

(

z

)

=

lim

ζ

→

z

⁡

f

♯

⁢

(

ζ

)

.

The second

definition

makes sense since a meromorphic functions has
only isolated poles, and thus

f

♯

⁢

(

ζ

)

is defined by the first equation when we are close to

z

. Some basic

properties

of the spherical derivative
are as follows.

Proposition

.

If

f

:

G

→

C

is a meromorphic function
then

•

f

♯

is a

continuous function

,

•

f

♯

⁢

(

z

)

<

∞

for all

z

∈

G

.

Note that sometimes the spherical derivative is also denoted
as

μ

⁢

(

f

)

⁢

(

z

)

rather then

f

♯

⁢

(

z

)

.

References

1

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

2

Theodore B. Gamelin.
.
Springer-Verlag, New York, New York, 2001.

Title

spherical derivative

Canonical

name

SphericalDerivative

Date of creation

2013-03-22 14:18:36

Last modified on

2013-03-22 14:18:36

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30D30

Generated on Fri Feb 9 20:19:18 2018 by

LaTeXML
table of partial fraction expansions

table of partial fraction expansions

The purpose of this entry is to collect a table of Mittag-Leffler type
partial fraction expansions for various functions.

1

Elementary Functions

π

⁢

cot

⁡

(

π

⁢

z

)

=

1

z

+

∑

n

=

1

∞

(

1

z

-

n

+

1

z

+

n

)

(1)

π

⁢

sec

⁡

(

π

⁢

z

)

=

2

1

-

2

⁢

z

+

∑

k

=

1

∞

(

-

1

)

k

+

1

⁢

(

2

2

⁢

k

-

1

-

2

⁢

z

-

2

2

⁢

k

+

1

-

2

⁢

z

)

(2)

2

Hypergeometric Functions

F

1

2

⁢

(

z

,

1

;

z

+

1

;

w

)

=

∑

k

=

0

∞

w

k

z

+

k

(4)

3

Gamma Functions

ψ

⁢

(

z

)

=

Γ

′

⁢

(

z

)

Γ

⁢

(

z

)

+

γ

=

1

z

+

∑

k

=

1

∞

(

1

k

-

1

z

+

k

)

(6)

(

-

1

)

n

⁢

ψ

(

n

)

⁢

(

z

)

n

!

=

∑

k

=

0

∞

1

(

z

+

k

)

n

(7)

Γ

⁢

(

x

)

⁢

Γ

⁢

(

1

2

)

Γ

⁢

(

x

+

1

2

)

=

∑

n

=

0

∞

(

2

⁢

n

)

!

2

2

⁢

n

⁢

(

n

!

)

2

⁢

1

x

+

n

(8)

Here

γ

is Mascheroni’s constant.

4

Elliptic Functions

℘

(

z

|

1

2

ω

,

1

2

ω

′

)

=

1

z

2

+

∑

|

k

|

+

|

k

′

|

≠

0

(

1

(

z

-

k

⁢

ω

-

k

′

⁢

ω

′

)

2

-

1

(

k

⁢

ω

+

k

′

⁢

ω

′

)

2

)

(10)

Title

table of partial fraction expansions

Canonical name

TableOfPartialFractionExpansions

Date of creation

2013-03-22 15:44:29

Last modified on

2013-03-22 15:44:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

20

Author

rspuzio (6075)

Entry type

Example

Classification

msc 30D30

Related topic

ElementaryFunction

Related topic

GammaFunction

Related topic

HypergeometricFunction

Related topic

EllipticFunction

Generated on Fri Feb 9 20:19:21 2018 by

LaTeXML
technique for computing residues

technique for computing residues

The following two facts are quite useful for computing

residues

:

If

f

has a pole of order at most

n

+

1

at

x

, then

Res

⁡

(

f

;

x

)

=

lim

y

→

x

⁡

1

n

!

⁢

d

n

d

⁢

y

n

⁢

(

(

y

-

x

)

n

+

1

⁢

f

⁢

(

y

)

)

.

If

g

is

regular

at

x

and

f

has a

simple pole

at

x

, then

Res

⁡

(

f

⁢

g

;

x

)

=

g

⁢

(

x

)

⁢

Res

⁡

(

f

;

x

)

.

Title

technique for computing residues

Canonical name

TechniqueForComputingResidues

Date of creation

2013-03-22 16:20:11

Last modified on

2013-03-22 16:20:11

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30D30

Related topic

CoefficientsOfLaurentSeries

Related topic

ResiduesOfTangentAndCotangent

Generated on Fri Feb 9 20:19:22 2018 by

LaTeXML
values of gamma function for small positive real values

values of gamma function for small positive real values

The following table lists values of

Γ

⁢

(

x

)

for real

0

≤

x

<

10

in steps of

1

10

. If the table is correct, it should confirm for a real

nonnegative integer

n

the following equality:

Γ

⁢

(

n

)

=

(

n

-

1

)

!

, with the notable exception of

n

=

0

, for which the value is complex infinity. Generally speaking, for values of

x

in the relation

n

<

x

<

(

n

+

1

)

, the relation

Γ

⁢

(

n

)

<

Γ

⁢

(

x

)

<

Γ

⁢

(

n

+

1

)

always holds, but not when

0

<

x

<

2

.

x

Γ

⁢

(

x

)

x

Γ

⁢

(

x

)

0

∞

5

24

0.1

9.51351

5.1

27.9318

0.2

4.59084

5.2

32.5781

0.3

2.99157

5.3

38.078

0.4

2.21816

5.4

44.5988

0.5

1.77245

5.5

52.3428

0.6

1.48919

5.6

61.5539

0.7

1.29806

5.7

72.5276

0.8

1.16423

5.8

85.6217

0.9

1.06863

5.9

101.27

1

1

6

120

1.1

0.951351

6.1

142.452

1.2

0.918169

6.2

169.406

1.3

0.897471

6.3

201.813

1.4

0.887264

6.4

240.834

1.5

0.886227

6.5

287.885

1.6

0.893515

6.6

344.702

1.7

0.908639

6.7

413.408

1.8

0.931384

6.8

496.606

1.9

0.961766

6.9

597.494

2

1

7

720

2.1

1.04649

7.1

868.957

2.2

1.1018

7.2

1050.32

2.3

1.16671

7.3

1271.42

2.4

1.24217

7.4

1541.34

2.5

1.32934

7.5

1871.25

2.6

1.42962

7.6

2275.03

2.7

1.54469

7.7

2769.83

2.8

1.67649

7.8

3376.92

2.9

1.82736

7.9

4122.71

3

2

8

5040

3.1

2.19762

8.1

6169.59

3.2

2.42397

8.2

7562.29

3.3

2.68344

8.3

9281.39

3.4

2.98121

8.4

11405.9

3.5

3.32335

8.5

14034.4

3.6

3.71702

8.6

17290.2

3.7

4.17065

8.7

21327.7

3.8

4.69417

8.8

26340

3.9

5.29933

8.9

32569.4

4

6

9

40320

4.1

6.81262

9.1

49973.7

4.2

7.75669

9.2

62010.8

4.3

8.85534

9.3

77035.6

4.4

10.1361

9.4

95809.5

4.5

11.6317

9.5

119292

4.6

13.3813

9.6

148696

4.7

15.4314

9.7

185551

4.8

17.8379

9.8

231792

4.9

20.6674

9.9

289868

Title

values of gamma function for small positive real values

Canonical name

ValuesOfGammaFunctionForSmallPositiveRealValues

Date of creation

2013-03-22 17:48:50

Last modified on

2013-03-22 17:48:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 30D30

Classification

msc 33B15

Generated on Fri Feb 9 20:19:24 2018 by

LaTeXML
Möbius transformation

Möbius transformation

A

Möbius

transformation

is a

bijection

on the

extended complex plane

ℂ

∪

{

∞

}

given by

f

⁢

(

z

)

=

{

a

c

if

⁢

z

=

∞

∞

if

⁢

z

=

-

d

c

a

⁢

z

+

b

c

⁢

z

+

d

otherwise

where

a

,

b

,

c

,

d

∈

ℂ

and

a

⁢

d

-

b

⁢

c

≠

0

It can be shown that the

inverse

, and

composition

of two Möbius transformations are similarly defined, and so the Möbius transformations form a group under composition.

The geometric

interpretation

of the Möbius group is that it is the group of

automorphisms

of the Riemann sphere.

Any Möbius map can be composed from the

elementary

transformations -

dilations

, translations and

inversions

. If we define a line to be a circle

passing through

∞

then it can be shown that a Möbius transformation maps circles to circles, by looking at each elementary transformation.

Title

Möbius transformation

Canonical name

MobiusTransformation

Date of creation

2013-03-22 12:23:19

Last modified on

2013-03-22 12:23:19

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

21

Author

Koro (127)

Entry type

Definition

Classification

msc 30D99

Synonym

fractional linear transformation

Synonym

linear fractional transformation

Related topic

ProofOfConformalMobiusCircleMapTheorem

Related topic

AutomorphismsOfUnitDisk

Related topic

UnitDiskUpperHalfPlaneConformalEquivalenceTheorem

Related topic

InversionOfPlane

Generated on Fri Feb 9 20:19:30 2018 by

LaTeXML
two series arising from the alternating zeta function

two series arising from the alternating zeta function

The terms of the series defining the

alternating zeta function

η

(

s

)

:=

∑

n

=

1

∞

(

-

1

)

n

-

1

n

s

(

Re

s

>

0

)

,

a.k.a. the

Dirichlet eta function

, may be split into their real and

imaginary parts

:

1

n

s

=

e

-

i

⁢

b

⁢

ln

⁡

n

n

a

=

cos

⁡

(

b

⁢

ln

⁡

n

)

n

a

-

i

⁢

sin

⁡

(

b

⁢

ln

⁡

n

)

n

a

Here,

s

=

a

+

i

⁢

b

with real

a

and

b

.  It follows the equation

η

⁢

(

s

)

=

-

∑

n

=

1

∞

(

-

1

)

n

n

a

⁢

cos

⁡

(

b

⁢

ln

⁡

n

)

+

i

⁢

∑

n

=

1

∞

(

-

1

)

n

n

a

⁢

sin

⁡

(

b

⁢

ln

⁡

n

)

(1)

containing two

Dirichlet series

.

The alternating zeta function and the

Riemann zeta function

are

connected

by the relation

ζ

⁢

(

s

)

=

η

⁢

(

s

)

1

-

2

1

-

s

(see the parent entry (

http://planetmath.org/AnalyticContinuationOfRiemannZetaToCriticalStrip

)).  The following conjecture concerning the above

real part series and imaginary part series

of (1) has been proved by Sondow [1] to be

equivalent

with the

Riemann hypothesis

.

Conjecture.

If the equations

∑

n

=

1

∞

(

-

1

)

n

n

a

⁢

cos

⁡

(

b

⁢

ln

⁡

n

)

=

0

and

∑

n

=

1

∞

(

-

1

)

n

n

a

⁢

sin

⁡

(

b

⁢

ln

⁡

n

)

=

0

are true for some pair of

real numbers

a

and

b

, then

a

=

1

/

2

or

a

=

1

.

References

1

Jonathan Sondow

: A simple counterexample to Havil’s “reformulation” of the Riemann hypothesis.  –

Elemente der Mathematik

67

(2012) 61–67.  Also available

http://arxiv.org/pdf/0706.2840v3.pdf

here.

Title

two series arising from the alternating zeta function

Canonical name

TwoSeriesArisingFromTheAlternatingZetaFunction

Date of creation

2013-06-06 19:13:30

Last modified on

2013-06-06 19:13:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Conjecture

Classification

msc 30D99

Classification

msc 30B50

Classification

msc 11M41

Synonym

trigonometric series

conjecture equivalent to the Riemann hypothesis

Related topic

EulerRelation

Generated on Fri Feb 9 20:19:41 2018 by

LaTeXML
Mergelyan’s theorem

Mergelyan’s theorem

Theorem

(Mergelyan).

Let

K

⊂

C

be a

compact subset

of the complex plane such that

C

\

K

(the complement of

K

) is

connected

, and let

f

:

K

→

C

be a

continuous function

which is also holomorphic
on the

interior

of

K

.

Then

f

is the uniform limit on

K

of holomorphic

polynomials

(polynomials in one complex variable).

So for any

ϵ

>

0

one can find a polynomial

p

⁢

(

z

)

=

∑

j

=

1

n

a

j

⁢

z

j

such that

|

f

⁢

(

z

)

-

p

⁢

(

z

)

|

<

ϵ

for all

z

∈

K

.

Do note that this theorem is not a weaker version of Runge’s theorem. Here, we do not
need

f

to be holomorphic on a

neighbourhood

of

K

,

but just on the interior of

K

.

For example, if the interior of

K

is empty, then

f

just needs to be

continuous

on

K

.

Further, it could be that the

closure

of the interior of

K

might not be all of

K

.

Consider

K

=

D

∪

[

-

10

,

10

]

,

where

D

is the closed

unit disc

. Then

K

has two lines coming out of either end of the disc and

f

needs to only be continuous there.

Also note that this theorem is distinct from the

Stone-Weierstrass theorem

. The point here is that the polynomials are
holomorphic in Mergelyan’s theorem.

References

1

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

2

Walter Rudin.
.
McGraw-Hill, Boston, Massachusetts, 1987.

Title

Mergelyan’s theorem

Canonical name

MergelyansTheorem

Date of creation

2013-03-22 14:23:59

Last modified on

2013-03-22 14:23:59

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30E10

Related topic

RungesTheorem

Generated on Fri Feb 9 20:19:43 2018 by

LaTeXML
Runge’s theorem

Runge’s theorem

Let

K

be a

compact subset

of

ℂ

, and let

E

be a subset of

ℂ

∞

=

ℂ

∪

{

∞

}

(the

extended complex plane

) which intersects every

connected component

of

ℂ

∞

-

K

. If

f

is an

analytic function

in an

open set

containing

K

, given

ε

>

0

, there is a

rational function

R

⁢

(

z

)

whose only poles are in

E

, such that

|

f

⁢

(

z

)

-

R

⁢

(

z

)

|

<

ε

for all

z

∈

K

.

Title

Runge’s theorem

Canonical name

RungesTheorem

Date of creation

2013-03-22 13:15:12

Last modified on

2013-03-22 13:15:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 30E10

Related topic

MergelyansTheorem

Generated on Fri Feb 9 20:19:45 2018 by

LaTeXML
absolute convergence implies convergence for an infinite product

absolute convergence implies convergence for an infinite product

If an infinite product is

absolutely convergent

(

http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct

) then it is convergent.

Title

absolute convergence

implies

convergence

for an infinite product

Canonical name

AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct

Date of creation

2013-03-22 13:36:03

Last modified on

2013-03-22 13:36:03

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 30E20

Related topic

AbsoluteConvergenceOfInfiniteProduct

Related topic

ConvergenceForAnInfiniteProduct

Related topic

AbsoluteConvergentInfiniteProductConverges

Generated on Fri Feb 9 20:19:46 2018 by

LaTeXML
absolute convergence of infinite product

absolute convergence of infinite product

An infinite product

∏

n

=

1

∞

(

1

+

a

n

)

is said to be

if

∏

n

=

1

∞

(

1

+

|

a

n

|

)

converges.

Title

absolute convergence of infinite product

Canonical

name

AbsoluteConvergenceOfInfiniteProduct

Date of creation

2013-03-22 13:36:00

Last modified on

2013-03-22 13:36:00

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 30E20

Related topic

AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct

Generated on Fri Feb 9 20:19:48 2018 by

LaTeXML