aarjaneiro/planetmath huggingface dataset:
regular at infinity

regular at infinity

When the

function

w

of one complex variable is

regular

in the

annulus

ϱ

<

|

z

|

<

∞

,

it has a

Laurent expansion

w

⁢

(

z

)

=

∑

n

=

-

∞

∞

c

n

⁢

z

n

.

(1)

If especially the coefficients

c

1

,

c

2

,

…

vanish, then we have

w

⁢

(

z

)

=

c

0

+

c

-

1

z

+

c

-

2

z

2

+

…

Using the

inversion

z

=

1

ζ

,  we see that the function

w

⁢

(

1

ζ

)

=

c

0

+

c

-

1

⁢

ζ

+

c

-

2

⁢

ζ

2

+

…

is regular in the disc

|

ζ

|

<

ϱ

.  Accordingly we can define that the function

w

is

regular at infinity

also.

For example,

w

⁢

(

z

)

:=

1

z

is regular at the point

z

=

∞

and

w

⁢

(

∞

)

=

0

.  Similarly,

e

1

z

is regular at

∞

and has there the value 1.

Title

regular at infinity

Canonical

name

RegularAtInfinity

Date of creation

2013-03-22 17:37:30

Last modified on

2013-03-22 17:37:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D20

Classification

msc 32A10

Synonym

analytic at infinity

Related topic

RegularFunction

Related topic

ClosedComplexPlane

Related topic

VanishAtInfinity

Related topic

ResidueAtInfinity

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

LaTeXML
analytic continuation of gamma function

analytic continuation of gamma function

The last of the parent entry (

http://planetmath.org/GammaFunction

) may be expressed as

Γ

⁢

(

z

)

=

Γ

⁢

(

z

+

n

)

z

⁢

(

z

+

1

)

⁢

(

z

+

2

)

⁢

⋯

⁢

(

z

+

n

-

1

)

.

(1)

According to the standard definition

Γ

⁢

(

z

)

:=

∫

0

∞

e

-

t

⁢

t

z

-

1

⁢

𝑑

t

,

the left hand side of (1) is defined only in the

right half-plane

ℜ

⁡

z

>

0

,
whereas the expression

Γ

⁢

(

z

+

n

)

is defined and

holomorphic

for

ℜ

⁡

z

>

-

n

and thus the right hand side of (1) is holomorphic in the half-plane

ℜ

⁡

z

>

-

n

except the points

0

,

-

1

,

-

2

,

…

,

-

(

n

-

1

)

where it has the poles of order 1.  Because the both sides of (1) are equal for

ℜ

⁡

z

>

0

,  the left side of (1) is the

analytic continuation

of

Γ

⁢

(

z

)

to
the half-plane

ℜ

⁡

z

>

-

n

.  And since the

positive integer

n

can be chosen
arbitrarily, the Euler’s

Γ

-

function

has been defined analytically to the whole

complex plane

.

Accordingly, the

gamma function

is unambiguous and holomorphic everywhere in

ℂ

except in the points

0

,

-

1

,

-

2

,

-

3

,

…

(2)

which are poles of order 1 of the function.  Hence,

Γ

⁢

(

z

)

is a

meromorphic function

.

For determining the

residue

of the function in the points (2), we rewrite the equation (1) as

Γ

⁢

(

z

)

=

Γ

⁢

(

z

+

n

+

1

)

z

⁢

(

z

+

1

)

⁢

(

z

+

2

)

⁢

⋯

⁢

(

z

+

n

)

.

In the point

z

=

-

n

we have

Γ

⁢

(

z

+

n

+

1

)

=

Γ

⁢

(

1

)

=

0

!

=

1

,

which implies (see the rule in the entry

coefficients of Laurent series

) that

Res

⁡

(

Γ

;

-

n

)

=

(

-

1

)

n

n

!

.

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title

analytic continuation of gamma function

Canonical name

AnalyticContinuationOfGammaFunction

Date of creation

2013-03-22 17:03:07

Last modified on

2013-03-22 17:03:07

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Derivation

Classification

msc 30D30

Classification

msc 30B40

Classification

msc 33B15

Synonym

residues of gamma function

Related topic

AnalyticContinuation

Related topic

EmptyProduct

Related topic

ResiduesOfTangentAndCotangent

Related topic

RolfNevanlinna

Generated on Fri Feb 9 20:18:21 2018 by

LaTeXML
Casorati-Weierstrass theorem

Casorati-Weierstrass theorem

Given a domain

U

⊂

ℂ

,

a

∈

U

, and

f

:

U

∖

{

a

}

→

ℂ

being

holomorphic

, then

a

is an

essential singularity

of

f

if and only if the image of any punctured

neighborhood

of

a

under

f

is dense in

ℂ

. Put another way, a

holomorphic function

can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any

complex

value.

Title

Casorati-Weierstrass theorem

Canonical name

CasoratiWeierstrassTheorem

Date of creation

2013-03-22 13:32:36

Last modified on

2013-03-22 13:32:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 30D30

Synonym

Weierstrass-Casorati theorem

Related topic

PicardsTheorem

Generated on Fri Feb 9 20:18:25 2018 by

LaTeXML
digamma and polygamma function

digamma and polygamma function

The

digamma function

is defined as the

logarithmic derivative

of the

gamma function

:

ψ

⁢

(

z

)

=

d

d

⁢

z

⁢

log

⁡

Γ

⁢

(

z

)

=

Γ

′

⁢

(

z

)

Γ

⁢

(

z

)

.

Likewise the

polygamma functions

are defined as higher order logarithmic derivatives of the gamma function:

ψ

(

n

)

⁢

(

z

)

=

d

n

d

⁢

z

n

⁢

log

⁡

Γ

⁢

(

z

)

.

These equations enjoy

functional equations

which are closely related to those of the gamma function:

ψ

⁢

(

z

+

1

)

=

ψ

⁢

(

z

)

+

1

z

ψ

⁢

(

1

-

z

)

=

ψ

⁢

(

z

)

+

π

⁢

cot

⁡

π

⁢

z

ψ

⁢

(

2

⁢

z

)

=

1

2

⁢

ψ

⁢

(

z

)

+

1

2

⁢

ψ

⁢

(

z

+

1

2

)

+

log

⁡

2

ψ

(

n

)

⁢

(

z

+

1

)

=

ψ

(

n

)

⁢

(

z

)

+

(

-

1

)

n

⁢

n

!

z

n

+

1

These

functions

have poles at the negative integers and can be expressed as

partial fraction

series:

ψ

⁢

(

z

)

=

-

γ

-

1

z

+

∑

k

=

1

∞

(

1

k

-

1

z

+

k

)

,

(1)

ψ

(

n

)

⁢

(

z

)

=

(

-

1

)

n

⁢

n

!

⁢

∑

k

=

0

∞

1

(

z

+

k

)

n

(2)

Here,

γ

is Euler–Mascheroni constant (

http://planetmath.org/EulerMascheroniConstant

).  Substituting

z

=

1

to (1), one gets the value

Γ

′

⁢

(

1

)

=

-

γ

.

Title

digamma and polygamma function

Canonical name

DigammaAndPolygammaFunction

Date of creation

2013-03-22 15:53:21

Last modified on

2013-03-22 15:53:21

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30D30

Classification

msc 33B15

Defines

digamma function

Defines

polygamma function

Generated on Fri Feb 9 20:18:27 2018 by

LaTeXML
dilogarithm function

dilogarithm function

The

dilogarithm function

Li

2

(

x

)

=

:

∑

n

=

1

∞

x

n

n

2

,

(1)

studied already by Leibniz, is a special case of the

polylogarithm function

Li

s

(

x

)

=

:

∑

n

=

1

∞

x

n

n

s

.

The

radius of convergence

of the series (1) is 1, whence the definition (1) is valid also in the

unit disc

of the

complex plane

.  For

0

≤

x

≤

1

,  the equation (1) is apparently equivalent to

Li

2

(

x

)

=

:

-

∫

0

x

ln

⁡

(

1

-

t

)

t

d

t

,

(2)

(cf.

logarithm series

of

ln

⁡

(

1

-

x

)

).  The

analytic continuation

of

Li

2

for

|

z

|

≥

1

can be made by

Li

2

(

z

)

=

:

-

∫

0

z

log

⁡

(

1

-

t

)

t

d

t

.

(3)

Thus

Li

2

⁢

(

z

)

is a multivalued

analytic function

of

z

.  Its

principal branch

is single-valued and is got by taking the principal branch of the

complex logarithm

; then

z

∈

ℂ

∖

[

1

,

∞

[

,

0

<

arg

(

z

-

1

)

<

2

π

.

For real values of

x

we have

Im

⁢

(

Li

2

⁢

(

x

)

)

=

{

0

for

⁢

x

≤

1

,

-

π

⁢

ln

⁡

x

⁢

for

⁢

x

>

1

.

According to (2), the

derivative

of the

dilogarithm

is

Li

2

′

⁢

(

x

)

=

-

ln

⁡

(

1

-

x

)

x

.

In terms of the

Bernoulli numbers

, the dilogarithm function has a

series expansion

more rapidly converging than (1):

Li

2

(

x

)

=

∑

n

=

1

∞

B

n

-

1

(

-

ln

⁡

(

1

-

x

)

)

n

n

!

(

|

ln

(

1

-

x

)

|

<

2

π

)

(4)

Some functional equations and values

Li

2

⁢

(

z

)

+

Li

2

⁢

(

-

z

)

=

1

2

⁢

Li

2

⁢

(

z

2

)

,

Li

2

⁢

(

z

)

+

Li

2

⁢

(

1

z

)

=

-

1

2

⁢

(

log

⁡

(

-

z

)

)

2

-

π

2

6

,

Li

2

⁢

(

i

⁢

z

)

-

i

⁢

Li

2

⁢

(

z

)

=

1

4

⁢

Li

2

⁢

(

-

z

2

)

,

Li

2

⁢

(

1

)

=

π

2

6

,

Li

2

⁢

(

2

)

=

π

2

4

-

i

⁢

π

⁢

ln

⁡

2

,

Li

2

⁢

(

i

)

=

-

π

2

48

-

i

⁢

G

Here,

G

is Catalan’s constant.

References

1

Anatol N. Kirillov

:

Dilogarithm identities

(1994). Available

http://arxiv.org/pdf/hep-th/9408113v2.pdf

here.

2

Leonard C. Maximon

: The dilogarithm function for complex argument.  –

Proc. R. Soc. Lond. A

459

(2003) 2807–2819.

Title

dilogarithm function

Canonical name

DilogarithmFunction

Date of creation

2013-03-22 19:34:58

Last modified on

2013-03-22 19:34:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D30

Classification

msc 33B15

Synonym

Spence’s

function

Related topic

ApplicationOfLogarithmSeries

Defines

polylogarithm function

Generated on Fri Feb 9 20:18:30 2018 by

LaTeXML
essential singularity

essential singularity

Let

U

⊂

ℂ

be a domain,

a

∈

U

, and let

f

:

U

∖

{

a

}

→

ℂ

be

holomorphic

. If the

Laurent series

expansion of

f

⁢

(

z

)

around

a

contains infinitely many terms with negative powers of

z

-

a

, then

a

is said to be an

essential singularity

of

f

. Any

singularity

of

f

is a

removable singularity

, a pole or an essential singularity.

If

a

is an essential singularity of

f

, then the image of any punctured

neighborhood

of

a

under

f

is dense in

ℂ

(the

Casorati-Weierstrass theorem

). In fact, an even stronger statement is true: according to Picard’s theorem, the image of any punctured neighborhood of

a

is

ℂ

, with the possible exception of a single point.

Title

essential singularity

Canonical name

EssentialSingularity

Date of creation

2013-03-22 13:32:10

Last modified on

2013-03-22 13:32:10

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

7

Author

pbruin (1001)

Entry type

Definition

Classification

msc 30D30

Related topic

LaurentSeries

Related topic

Pole

Related topic

RemovableSingularity

Related topic

PicardsTheorem

Related topic

RiemannsRemovableSingularityTheorem

Generated on Fri Feb 9 20:18:32 2018 by

LaTeXML
Euler reflection formula

Euler reflection formula

Theorem 1

(

Euler Reflection Formula

)

Γ

⁢

(

x

)

⁢

Γ

⁢

(

1

-

x

)

=

π

sin

⁡

(

π

⁢

x

)

Proof:

We have

1

Γ

⁢

(

x

)

=

x

⁢

e

γ

⁢

x

⁢

∏

n

=

1

∞

(

(

1

+

x

n

)

⁢

e

-

x

/

n

)

and thus

1

Γ

⁢

(

x

)

⁢

1

Γ

⁢

(

-

x

)

=

-

x

2

⁢

e

γ

⁢

x

⁢

e

-

γ

⁢

x

⁢

∏

n

=

1

∞

(

(

1

+

x

n

)

⁢

e

-

x

/

n

)

⁢

(

(

1

-

x

n

)

⁢

e

x

/

n

)

=

-

x

2

⁢

∏

n

=

1

∞

(

1

-

x

2

n

2

)

But

Γ

⁢

(

1

-

x

)

=

-

x

⁢

Γ

⁢

(

-

x

)

and thus

1

Γ

⁢

(

x

)

⁢

1

Γ

⁢

(

1

-

x

)

=

x

⁢

∏

n

=

1

∞

(

1

-

x

2

n

2

)

Now, using the formula (

http://planetmath.org/ExamplesOfInfiniteProducts

) for

sin

⁡

x

/

x

, we have

sin

⁡

(

π

⁢

x

)

=

π

⁢

x

⁢

∏

n

=

1

∞

(

1

-

x

2

n

2

)

so that

1

Γ

⁢

(

x

)

⁢

1

Γ

⁢

(

1

-

x

)

=

sin

⁡

(

π

⁢

x

)

π

and the result follows.

Title

Euler reflection formula

Canonical name

EulerReflectionFormula

Date of creation

2013-03-22 16:23:37

Last modified on

2013-03-22 16:23:37

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 30D30

Classification

msc 33B15

Generated on Fri Feb 9 20:18:34 2018 by

LaTeXML
evaluating the gamma function at 1/2

evaluating the gamma function at 1/2

In the entry on the

gamma function

it is mentioned that

Γ

⁢

(

1

/

2

)

=

π

. In this entry we reduce the proof of this claim to the
problem of computing the area under the bell curve. First note that by
definition of the gamma function,

Γ

⁢

(

1

/

2

)

=

∫

0

∞

e

-

x

⁢

x

-

1

/

2

⁢

𝑑

x

=

2

⁢

∫

0

∞

e

-

x

⁢

1

2

⁢

x

⁢

𝑑

x

.

Performing the substitution

u

=

x

, we find that

d

⁢

u

=

1

2

⁢

x

⁢

d

⁢

x

, so

Γ

⁢

(

1

/

2

)

=

2

⁢

∫

0

∞

e

-

u

2

⁢

𝑑

u

=

∫

ℝ

e

-

u

2

⁢

𝑑

u

,

where the last equality holds because

e

-

u

2

is an

even function

.
Since the area under the bell curve is

π

, it follows that

Γ

⁢

(

1

/

2

)

=

π

.

Title

evaluating the gamma function at 1/2

Canonical name

EvaluatingTheGammaFunctionAt12

Date of creation

2013-03-22 16:57:13

Last modified on

2013-03-22 16:57:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 30D30

Classification

msc 33B15

Related topic

AreaUnderGaussianCurve

Related topic

LaplaceTransformOfPowerFunction

Generated on Fri Feb 9 20:18:36 2018 by

LaTeXML
Hankel contour integral

Hankel contour integral

Hankel’s

contour integral

is a unit (and nilpotent) for

gamma function

over

ℂ

. That is,

(

i

2

⁢

π

⁢

∫

𝒞

(

-

t

)

-

z

⁢

e

-

t

⁢

𝑑

t

)

⁢

Γ

⁢

(

z

)

=

1

,

|

z

|

<

∞

.

Hankel’s

integral

is

holomorphic

with simple zeros in

ℤ

≤

0

. Its path of integration starts on the positive real axis

ad infinitum

, rounds the origin counterclockwise and returns to

+

∞

. As an example of application of Hankel’s integral, we have

i

2

⁢

π

⁢

∫

𝒞

(

-

t

)

-

1

2

⁢

e

-

t

⁢

𝑑

t

=

1

π

,

where the path of integration is the one above mentioned.

Title

Hankel contour integral

Canonical

name

HankelContourIntegral

Date of creation

2013-03-22 17:27:50

Last modified on

2013-03-22 17:27:50

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Result

Classification

msc 30D30

Classification

msc 33B15

Generated on Fri Feb 9 20:18:43 2018 by

LaTeXML
incomplete gamma function

incomplete gamma function

The

incomplete gamma function

is defined as the

indefinite integral

of the integrand of gamma

integral

. There are several definitions which differ in details of normalization and constant of integration:

γ

⁢

(

a

,

x

)

=

∫

0

x

e

-

t

⁢

t

a

-

1

⁢

𝑑

t

Γ

⁢

(

a

,

x

)

=

∫

x

∞

e

-

t

⁢

t

a

-

1

⁢

𝑑

t

=

Γ

⁢

(

a

)

-

γ

⁢

(

a

,

x

)

P

⁢

(

a

,

x

)

=

1

Γ

⁢

(

a

)

⁢

∫

0

x

e

-

t

⁢

t

a

-

1

⁢

𝑑

t

=

γ

⁢

(

a

,

x

)

Γ

⁢

(

a

)

γ

*

⁢

(

a

,

x

)

=

x

-

a

Γ

⁢

(

a

)

⁢

∫

0

x

e

-

t

⁢

t

a

-

1

⁢

𝑑

t

=

γ

⁢

(

a

,

x

)

x

a

⁢

Γ

⁢

(

a

)

I

⁢

(

a

,

x

)

=

1

Γ

⁢

(

a

+

1

)

⁢

∫

0

x

⁢

a

+

1

e

-

t

⁢

t

a

⁢

𝑑

t

=

γ

⁢

(

a

+

1

,

x

⁢

a

+

1

)

Γ

⁢

(

a

+

1

)

C

⁢

(

a

,

x

)

=

∫

x

∞

t

a

-

1

⁢

cos

⁡

t

⁢

d

⁢

t

S

⁢

(

a

,

x

)

=

∫

x

∞

t

a

-

1

⁢

sin

⁡

t

⁢

d

⁢

t

E

n

⁢

(

x

)

=

∫

1

∞

e

-

x

⁢

t

⁢

t

-

n

⁢

𝑑

t

=

x

n

-

1

⁢

Γ

⁢

(

1

-

n

)

-

x

n

-

1

⁢

γ

⁢

(

1

-

n

,

x

)

α

n

⁢

(

x

)

=

∫

1

∞

e

-

x

⁢

t

⁢

t

n

⁢

𝑑

t

=

x

-

n

-

1

⁢

Γ

⁢

(

1

+

n

)

-

x

-

n

-

1

⁢

γ

⁢

(

1

+

n

,

x

)

For convenience of translating notations, these variants have been expressed in terms of

γ

.

Title

incomplete gamma function

Canonical name

IncompleteGammaFunction

Date of creation

2013-03-22 15:36:48

Last modified on

2013-03-22 15:36:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30D30

Classification

msc 33B15

Related topic

SineIntegralInInfinity

Generated on Fri Feb 9 20:18:46 2018 by

LaTeXML
incomplete gamma function recurrence formula

incomplete gamma function recurrence formula

The

incomplete gamma function

satisfies the following recurrence formula:

γ

⁢

(

a

+

1

,

x

)

=

a

⁢

γ

⁢

(

a

,

x

)

-

x

a

⁢

e

-

x

This can be derived by

integration by parts

:

∫

0

x

e

-

t

⁢

t

a

⁢

𝑑

t

=

-

∫

0

x

t

a

⁢

𝑑

e

-

t

=

a

⁢

∫

0

x

t

a

-

1

⁢

𝑑

e

-

t

-

x

a

⁢

e

-

x

In terms of other variants of the incomplete gamma function, the recursion relation appears as follows:

P

⁢

(

a

+

1

,

x

)

=

P

⁢

(

a

,

x

)

-

x

a

⁢

e

-

x

Γ

⁢

(

a

+

1

)

γ

*

⁢

(

a

-

1

,

x

)

=

x

⁢

γ

*

⁢

(

a

,

x

)

+

e

-

x

Γ

⁢

(

a

)

Title

incomplete gamma function recurrence formula

Canonical name

IncompleteGammaFunctionRecurrenceFormula

Date of creation

2013-03-22 15:36:50

Last modified on

2013-03-22 15:36:50

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30D30

Classification

msc 33B15

Generated on Fri Feb 9 20:18:48 2018 by

LaTeXML
Marty’s theorem

Marty’s theorem

Theorem

(Marty).

A set

F

of

meromorphic functions

is a

normal family

on a domain

G

⊂

C

if and only if the

spherical
derivatives

are uniformly bounded (uniformly over

F

)
on each

compact subset

of

G

.

Here normal convergence (convergence on compact subsets) is given using the

spherical metric

and not the

standard metric

of the

complex plane

. That is, if

σ

is the spherical metric then we will say

f

n

→

f

normally
if

σ

⁢

(

f

n

⁢

(

z

)

,

f

⁢

(

z

)

)

converges

to 0 uniformly on compact subsets.

A related theorem can be stated.

Theorem

.

If

f

n

⁢

(

z

)

→

f

⁢

(

z

)

uniformly in the spherical metric on compact subsets of

G

⊂

C

then

f

n

♯

⁢

(

z

)

→

f

♯

⁢

(

z

)

uniformly
on compact subsets of

G

.

Here

f

♯

denotes the spherical derivative of

f

.

References

1

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

Title

Marty’s theorem

Canonical name

MartysTheorem

Date of creation

2013-03-22 14:18:39

Last modified on

2013-03-22 14:18:39

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30D30

Generated on Fri Feb 9 20:18:50 2018 by

LaTeXML
meromorphic

meromorphic

Let

U

⊂

ℂ

be a domain. A

function

f

:

U

→

ℂ

is

meromorphic

if

f

is

holomorphic

except at an

isolated set

of poles.

It can be proven that if

f

is meromorphic then its set of poles does not have an

accumulation point

.

Title

meromorphic

Canonical

name

Meromorphic

Date of creation

2013-03-22 12:05:53

Last modified on

2013-03-22 12:05:53

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 30D30

Generated on Fri Feb 9 20:18:51 2018 by

LaTeXML
meromorphic extension

meromorphic extension

Let

A

⊂

B

⊆

ℂ

and

f

:

A

→

ℂ

be

analytic

. A

meromorphic extension

of

f

is a

meromorphic function

g

:

B

→

ℂ

such that

g

|

A

=

f

.

The meromorphic extension of an

analytic function

to a larger domain (

http://planetmath.org/Domain

) is unique; i.e. (

http://planetmath.org/Ie

), using the above notation, if

h

:

B

→

ℂ

has the property that

h

|

A

=

f

, then

g

=

h

on

B

.

Occasionally, an analytic function and its meromorphic extension are denoted using the same notation. A common example of this phenomenon is the

Riemann zeta function

.

Title

meromorphic extension

Canonical name

MeromorphicExtension

Date of creation

2013-03-22 16:07:26

Last modified on

2013-03-22 16:07:26

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 30D30

Synonym

meromorphic continuation

Related topic

AnalyticContinuationOfRiemannZeta

Related topic

RestrictionOfAFunction

Generated on Fri Feb 9 20:18:53 2018 by

LaTeXML
Mittag-Leffler’s theorem

Mittag-Leffler’s theorem

Let

G

be an open subset of

ℂ

, let

{

a

k

}

be a

sequence

of distinct points in

G

which has no

limit point

in

G

. For each

k

, let

A

1

⁢

k

,

…

,

A

m

k

⁢

k

be arbitrary

complex

coefficients

, and define

S

k

⁢

(

z

)

=

∑

j

=

1

m

k

A

j

⁢

k

(

z

-

a

k

)

j

.

Then there exists a

meromorphic function

f

on

G

whose poles are exactly the points

{

a

k

}

and such that the singular part of

f

at

a

k

is

S

k

⁢

(

z

)

, for each

k

.

Title

Mittag-Leffler’s theorem

Canonical name

MittagLefflersTheorem

Date of creation

2013-03-22 13:15:15

Last modified on

2013-03-22 13:15:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Theorem

Classification

msc 30D30

Related topic

WeierstrassFactorizationTheorem

Generated on Fri Feb 9 20:18:54 2018 by

LaTeXML
pole

pole

Let

U

⊂

ℂ

be a domain and let

a

∈

ℂ

. A

function

f

:

U

→

ℂ

has a

pole

at

a

if it can be represented by a

Laurent series

centered about

a

with only finitely many terms of negative

exponent

; that is,

f

⁢

(

z

)

=

∑

k

=

-

n

∞

c

k

⁢

(

z

-

a

)

k

in some nonempty

deleted neighborhood

of

a

, with

c

-

n

≠

0

, for some

n

∈

ℕ

. The number

n

is called the

order

of the pole. A

simple pole

is a pole of order 1.

Title

pole

Canonical name

Pole

Date of creation

2013-03-22 12:05:56

Last modified on

2013-03-22 12:05:56

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 30D30

Related topic

EssentialSingularity

Defines

simple pole

Defines

simple

Generated on Fri Feb 9 20:18:56 2018 by

LaTeXML
proof of Casorati-Weierstrass theorem

proof of Casorati-Weierstrass theorem

Assume that

a

is an

essential singularity

of

f

. Let

V

⊂

U

be a punctured

neighborhood

of

a

, and let

λ

∈

ℂ

.
We have to show that

λ

is a limit point of

f

⁢

(

V

)

. Suppose it
is not, then there is an

ϵ

>

0

such that

|

f

⁢

(

z

)

-

λ

|

>

ϵ

for all

z

∈

V

, and the

function

g

:

V

→

ℂ

,

z

↦

1

f

⁢

(

z

)

-

λ

is

bounded

, since

|

g

⁢

(

z

)

|

=

1

|

f

⁢

(

z

)

-

λ

|

<

ϵ

-

1

for all

z

∈

V

. According to Riemann’s

removable singularity

theorem, this implies that

a

is a removable singularity of

g

, so
that

g

can be extended to a

holomorphic function

g

¯

:

V

∪

{

a

}

→

ℂ

. Now

f

⁢

(

z

)

=

1

g

¯

⁢

(

z

)

-

λ

for

z

≠

a

, and

a

is either a removable singularity of

f

(if

g

¯

⁢

(

z

)

≠

0

) or a pole of order

n

(if

g

¯

has a zero of
order

n

at

a

). This contradicts our assumption that

a

is an
essential singularity, which means that

λ

must be a limit
point of

f

⁢

(

V

)

. The

argument

holds for all

λ

∈

ℂ

,
so

f

⁢

(

V

)

is dense in

ℂ

for any punctured neighborhood

V

of

a

.

To prove the converse, assume that

f

⁢

(

V

)

is dense in

ℂ

for
any punctured neighborhood

V

of

a

. If

a

is a removable
singularity, then

f

is bounded near

a

, and if

a

is a pole,

f

⁢

(

z

)

→

∞

as

z

→

a

. Either of these possibilities
contradicts the assumption that the image of any punctured
neighborhood of

a

under

f

is dense in

ℂ

, so

a

must be
an essential singularity of

f

.

Title

proof of Casorati-Weierstrass theorem

Canonical name

ProofOfCasoratiWeierstrassTheorem

Date of creation

2013-03-22 13:32:40

Last modified on

2013-03-22 13:32:40

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Proof

Classification

msc 30D30

Related topic

PicardsTheorem

Generated on Fri Feb 9 20:18:58 2018 by

LaTeXML