aarjaneiro/planetmath huggingface dataset:
absolute convergence of infinite product and series

absolute convergence of infinite product and series

Theorem.

The infinite product

∏

n

=

1

∞

(

1

+

c

n

)

converges absolutely

(

http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct

) if and only if the series

∑

n

=

1

∞

c

n

with

complex

c

n

converges absolutely (

http://planetmath.org/AbsoluteConvergence

).

Proof.

The theorem follows directly from the theorems of the entries

absolutely convergent infinite product converges

and infinite product of sums

1

+

a

i

(

http://planetmath.org/InfiniteProductOfSums1a_i

).

Title

absolute convergence of infinite product and series

Canonical name

AbsoluteConvergenceOfInfiniteProductAndSeries

Date of creation

2013-03-22 18:41:18

Last modified on

2013-03-22 18:41:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Axiom

Classification

msc 30E20

Related topic

AbsoluteConvergentInfiniteProductConverges

Related topic

InfiniteProductOfSums1a_i

Related topic

OrderOfFactorsInInfiniteProduct

Generated on Fri Feb 9 20:19:49 2018 by

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argument principle

argument principle

If a

function

f

is

meromorphic

on the interior of a rectifiable

simple closed curve

C

, then

1

2

⁢

π

⁢

i

⁢

∮

C

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

(1)

equals the

difference

between the number of zeros and the number of poles of

f

counted with

multiplicity

. (For example, a zero of order two counts as two zeros; a pole of order three counts as three poles.)
This fact is known as the

argument principle

.

The principle may be stated in another form which makes the origin of the name apparent: If a function

f

is meromorphic on the interior of a rectifiable simple closed curve

C

and has

m

poles and

n

zeros on the interior of

C

, then the

argument

of

f

increases by

2

⁢

π

⁢

(

n

-

m

)

upon traversing

C

. The

relation

of this statement to the previous statement is easy to see. Note that

f

′

/

f

=

(

log

⁡

f

)

′

and that

log

⁡

(

z

)

=

log

⁡

|

z

|

+

i

⁢

arg

⁡

z

. Substituting this into

formula

(

1

), we find

2

⁢

π

⁢

i

⁢

(

n

-

m

)

=

∮

C

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

=

∮

C

d

⁢

log

⁡

|

f

⁢

(

z

)

|

+

i

⁢

∮

C

d

⁢

arg

⁡

(

f

⁢

(

z

)

)

.

The first integral on the rightmost side of this

equation

equals zero because

log

⁡

|

f

|

is

single-valued

. The second integral on the rightmost side equals the change in the argument as one traverses

C

. Cancelling the

i

from both sides, we conclude that the change in the argument equals

2

⁢

π

⁢

(

n

-

m

)

.

Note also that the integral (

1

)
is the

winding number

, about zero, of the image curve

f

∘

C

.

Title

argument principle

Canonical name

ArgumentPrinciple

Date of creation

2013-03-22 14:34:28

Last modified on

2013-03-22 14:34:28

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Algorithm

Classification

msc 30E20

Synonym

Cauchy’s argument principle

Defines

argument principle

Generated on Fri Feb 9 20:19:51 2018 by

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calculation of contour integral

calculation of contour integral

We will determine the important

complex integral

I

:=

∮

C

(

z

-

z

0

)

n

⁢

𝑑

z

where

C

is the circumference of the circle

|

z

-

z

0

|

=

ϱ

taken anticlockwise and

n

an arbitrary integer.

Let’s take the “direction angle” of the radius of

C

as the parametre

t

, i.e.

t

:=

arg

⁡

|

z

-

z

0

|

.

Then on

C

, we have

z

-

z

0

=

ϱ

⁢

e

i

⁢

t

,

0

≦

t

≦

2

⁢

π

and

d

⁢

z

=

i

⁢

ϱ

⁢

e

i

⁢

t

⁢

d

⁢

t

,

(

z

-

z

0

)

n

=

ϱ

n

⁢

e

i

⁢

n

⁢

t

,

whence

I

=

∫

0

2

⁢

π

ϱ

n

⁢

e

i

⁢

n

⁢

t

⁢

i

⁢

ϱ

⁢

e

i

⁢

t

⁢

𝑑

t

=

i

⁢

ϱ

n

+

1

⁢

∫

0

2

⁢

π

e

i

⁢

(

n

+

1

)

⁢

t

⁢

𝑑

t

.

In the case

n

=

-

1

one gets trivially

I

=

2

⁢

i

⁢

π

.  If

n

≠

-

1

,  we obtain

I

=

i

⁢

ϱ

n

+

1

⁢

/

t

=

0

2

⁢

π

⁡

e

i

⁢

(

n

+

1

)

⁢

t

i

⁢

(

n

+

1

)

=

ϱ

n

+

1

n

+

1

⁢

(

1

-

1

)

=

0

,

using the fact that

2

⁢

i

⁢

π

is a period of the

exponential function

(

http://planetmath.org/PeriodicityOfExponentialFunction

).

Hence we can write the result

∮

C

(

z

-

z

0

)

n

⁢

𝑑

z

=

{

2

⁢

i

⁢

π

if

⁢

n

=

-

1

,

0

if

⁢

n

∈

ℤ

∖

{

-

1

}

.

Title

calculation of contour integral

Canonical name

CalculationOfContourIntegral

Date of creation

2013-03-22 19:14:16

Last modified on

2013-03-22 19:14:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 30E20

Classification

msc 30A99

Related topic

AntiderivativeOfComplexFunction

Related topic

SubstitutionNotation

Related topic

ProofOfCauchyIntegralFormula

Generated on Fri Feb 9 20:19:53 2018 by

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Cauchy integral formula

Cauchy integral formula

The formulas.

Let

D

=

{

z

∈

ℂ

:

|

z

-

z

0

|

<

R

}

be an

open disk

in the

complex plane

, and let

f

⁢

(

z

)

be a

holomorphic

1

1

It is
necessary to draw a distinction between

holomorphic functions

(those having
a

complex derivative

) and

analytic functions

(those representable by

power series

). The two

concepts

are, in fact,

equivalent

, but
the standard proof of this fact uses the

Cauchy Integral Formula

with the (apparently) weaker holomorphicity

hypothesis

.

function

defined on some open domain that contains

D

and its

boundary

. Then,
for every

z

∈

D

we have

f

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

ζ

-

z

⁢

𝑑

ζ

f

′

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

(

ζ

-

z

)

2

⁢

𝑑

ζ

⋮

f

(

n

)

⁢

(

z

)

=

n

!

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

Here

C

=

∂

⁡

D

is the corresponding circular boundary

contour

,
oriented counterclockwise, with the most obvious parameterization
given by

ζ

=

z

0

+

R

⁢

e

i

⁢

t

,

0

≤

t

≤

2

⁢

π

.

Discussion.

The first of the above

formulas

underscores the “rigidity” of
holomorphic functions. Indeed, the values of the holomorphic function
inside a disk

D

are completely specified by its values on the
boundary of the disk. The second formula is useful, because it gives
the

derivative

in terms of an

integral

, rather than as the outcome of
a limit process.

Generalization.

The following technical

generalization

of the formula is needed for
the treatment of

removable singularities

. Let

S

be a finite subset
of

D

, and suppose that

f

⁢

(

z

)

is holomorphic for all

z

∉

S

, but also that

f

⁢

(

z

)

is

bounded

near all

z

∈

S

. Then, the
above formulas are valid for all

z

∈

D

∖

S

.

Using the

Cauchy residue theorem

, one can further generalize the integral formula to the situation where

D

is any domain and

C

is any closed rectifiable curve in

D

; in this case, the formula becomes

η

⁢

(

C

,

z

)

⁢

f

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

ζ

-

z

⁢

𝑑

ζ

where

η

⁢

(

C

,

z

)

denotes the

winding number

of

C

. It is valid for all points

z

∈

D

∖

S

which are not on the curve

C

.

Title

Cauchy integral formula

Canonical name

CauchyIntegralFormula

Date of creation

2013-03-22 12:04:46

Last modified on

2013-03-22 12:04:46

Owner

djao (24)

Last modified by

djao (24)

Numerical id

25

Author

djao (24)

Entry type

Theorem

Classification

msc 30E20

Generated on Fri Feb 9 20:19:56 2018 by

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Cauchy residue theorem

Cauchy residue theorem

Let

U

⊂

ℂ

be a

simply connected

domain, and suppose

f

is a

complex

valued

function

which is defined and

analytic

on all but finitely many points

a

1

,

…

,

a

m

of

U

. Let

C

be a

closed curve

in

U

which does not intersect any of the

a

i

. Then

∫

C

f

⁢

(

z

)

⁢

𝑑

z

=

2

⁢

π

⁢

i

⁢

∑

i

=

1

m

η

⁢

(

C

,

a

i

)

⁢

Res

⁡

(

f

;

a

i

)

,

where

η

⁢

(

C

,

a

i

)

:=

1

2

⁢

π

⁢

i

⁢

∫

C

d

⁢

z

z

-

a

i

is the

winding number

of

C

about

a

i

, and

Res

⁡

(

f

;

a

i

)

denotes the

residue

of

f

at

a

i

.

The

Cauchy residue theorem

generalizes both the

Cauchy integral theorem

(because

analytic functions

have no poles) and the

Cauchy integral formula

(because

f

⁢

(

x

)

/

(

x

-

a

)

n

for analytic

f

has exactly one pole at

x

=

a

with residue

Res

(

f

(

x

)

/

(

x

-

a

)

n

,

a

)

=

f

(

n

)

(

a

)

/

n

!

)

.

Title

Cauchy residue theorem

Canonical name

CauchyResidueTheorem

Date of creation

2013-03-22 12:04:58

Last modified on

2013-03-22 12:04:58

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Theorem

Classification

msc 30E20

Synonym

Cauchy residue formula

Synonym

residue theorem

Related topic

Residue

Related topic

CauchyIntegralFormula

Related topic

CauchyIntegralTheorem

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

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closed curve theorem

closed curve theorem

Let

U

⊂

ℂ

be a

simply connected

domain, and suppose

f

:

U

⟶

ℂ

is

holomorphic

. Then

∫

C

f

⁢

(

z

)

⁢

𝑑

z

=

0

for any smooth

closed curve

C

in

U

.

More generally, if

U

is any domain, and

C

1

and

C

2

are two

homotopic

smooth closed curves in

U

, then

∫

C

1

f

⁢

(

z

)

⁢

𝑑

z

=

∫

C

2

f

⁢

(

z

)

⁢

𝑑

z

.

for any

holomorphic function

f

:

U

⟶

ℂ

.

Title

closed curve theorem

Canonical name

ClosedCurveTheorem

Date of creation

2013-03-22 12:04:43

Last modified on

2013-03-22 12:04:43

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Theorem

Classification

msc 30E20

Related topic

CauchyIntegralTheorem

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

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conformal mapping

conformal mapping

A mapping

f

:

ℝ

m

→

ℝ

n

which preserves the magnitude and

orientation

of the angles between any two curves which intersect in a given point

z

0

is said to be

conformal

at

z

0

. A mapping that is conformal at any point in a domain

D

is said to be conformal in

D

.

An important special case is when

m

=

n

=

2

. In this case, we may identify

ℝ

2

with

ℂ

and speak of

conformal mappings

of the

complex plane

. It can be shown that a mapping

f

:

ℂ

→

ℂ

is conformal if and only if

f

is a

complex analytic function

.  The

complex conjugate

f

¯

of a conformal mapping

f

is

inversely conformal

, i.e. it preserves the magnitude but reverses the orientation of angles.

If

m

=

n

, then we can study invertible conformal mappings. It is clear from the definition that the

composition

of two such maps and the

inverse

of any such map is again an invertible conformal mapping, so the set of such mappings forms a group. In the case

m

=

n

>

2

, the group of conformal mappings is finite dimensional and is generated by

rotations

,

translations

, and spherical

inversions

.

This notion of conformal mappings can be generalized to any setting in which it makes sense to speak of angles between curves or angles between

tangent vectors

. In particular, one can consider conformal mappings of

Riemannian manifolds

. It can be shown that, if

(

M

,

g

)

and

(

N

,

h

)

are Riemannian manifolds, then a map

f

:

M

→

N

is conformal if and only if

f

*

⁢

h

=

s

⁢

g

for some scalar field

s

(on

M

).

Title

conformal mapping

Canonical name

ConformalMapping

Date of creation

2013-03-22 13:35:42

Last modified on

2013-03-22 13:35:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

19

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30E20

Classification

msc 53A30

Related topic

QuasiconformalMapping

Related topic

ConformalMappingTheorem

Related topic

AngleBetweenTwoLines

Related topic

SchwarzChristoffelTransformation

Related topic

CategoryOfRiemannianManifolds

Defines

conformal

Defines

inversely conformal

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

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conformal Möbius circle map theorem

conformal Möbius circle map theorem

Any conformal map that maps the interior of the

unit disc

onto itself is a Möbius

transformation

.

Title

conformal Möbius circle map

theorem

Canonical name

ConformalMobiusCircleMapTheorem

Date of creation

2013-03-22 13:35:52

Last modified on

2013-03-22 13:35:52

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

7

Author

Johan (1032)

Entry type

Theorem

Classification

msc 30E20

Related topic

UnitDisk

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

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convergence condition of infinite product

convergence condition of infinite product

Let us think the

sequence

u

1

,

u

1

⁢

u

2

,

u

1

⁢

u

2

⁢

u

3

,

…

In the

complex analysis

, one often uses the definition of the

convergence

of an

infinite product

∏

k

=

1

∞

u

k

where the case

lim

k

→

∞

⁡

u

1

⁢

u

2

⁢

…

⁢

u

k

=

0

is excluded.  Then one has the

Theorem

.

The infinite product

∏

k

=

1

∞

u

k

of the non-zero

complex numbers

u

1

,

u

2

, … is

convergent

iff for every positive number

ε

there exists a positive number

n

ε

such that the condition

|

u

n

+

1

⁢

u

n

+

2

⁢

…

⁢

u

n

+

p

-

1

|

<

ε

∀

p

∈

ℤ

+

is true as soon as

n

≧

n

ε

.

Corollary.

If the infinite product

converges

, then we necessarily have

lim

k

→

∞

⁡

u

k

=

1

. (Cf. the necessary condition of convergence of series.)

When the infinite product converges, we say that the

value of the infinite product

is equal to

lim

k

→

∞

⁡

u

1

⁢

u

2

⁢

…

⁢

u

k

.

Title

convergence condition of infinite product

Canonical name

ConvergenceConditionOfInfiniteProduct

Date of creation

2013-03-22 14:37:22

Last modified on

2013-03-22 14:37:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Related topic

OrderOfFactorsInInfiniteProduct

Related topic

NecessaryConditionOfConvergence

Defines

infinite product

Defines

value of infinite product

Generated on Fri Feb 9 20:20:07 2018 by

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convergence criterion for infinite product

convergence criterion for infinite product

Let

∏

n

=

1

∞

p

n

be an infinite product. We have the
following:

•

∏

n

=

1

∞

p

n

is

convergent

iff

∑

n

=

1

∞

ln

⁡

p

n

is convergent.

•

∏

n

=

1

∞

(

1

+

p

n

)

is convergent iff

∑

n

=

1

∞

p

n

converges absolutely.

Title

convergence criterion for infinite product

Canonical name

ConvergenceCriterionForInfiniteProduct

Date of creation

2013-03-22 14:50:07

Last modified on

2013-03-22 14:50:07

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

7

Author

aplant (12431)

Entry type

Theorem

Classification

msc 30E20

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

LaTeXML
convergence/divergence for an infinite product

convergence/divergence for an infinite product

Consider

∏

n

=

1

∞

p

n

. We say that this infinite product

converges

iff the finite products

P

m

=

∏

n

=

1

m

p

n

⟶

P

converge. Otherwise the infinite product is called

divergent

.

Title

convergence

/

divergence

for an infinite product

Canonical name

ConvergencedivergenceForAnInfiniteProduct

Date of creation

2013-03-22 13:36:11

Last modified on

2013-03-22 13:36:11

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

13

Author

aoh45 (5079)

Entry type

Definition

Classification

msc 30E20

Related topic

AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct

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

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estimating theorem of contour integral

estimating theorem of contour integral

Theorem

.

If

f

is a continuous

complex function

on the

rectifiable curve

γ

of the

complex plane

, then

|

∫

γ

f

⁢

(

z

)

⁢

𝑑

z

|

≦

max

z

∈

γ

⁡

|

f

⁢

(

z

)

|

⋅

l

,

(1)

where

l

=

∫

γ

|

d

⁢

z

|

is the of

γ

.

The form of (1) concerning the continuous real function

f

on the

interval

[

a

,

b

]

is

|

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

|

≦

max

a

≦

x

≦

b

⁡

|

f

⁢

(

x

)

|

⋅

(

b

-

a

)

.

For applications of this important theorem, see the example of using

residue theorem

.

Title

estimating theorem of contour integral

Canonical name

EstimatingTheoremOfContourIntegral

Date of creation

2013-03-22 15:19:36

Last modified on

2013-03-22 15:19:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Classification

msc 30A99

Synonym

estimation theorem of integral

Synonym

integral estimating theorem

Related topic

MinimalAndMaximalNumber

Related topic

AnalyticContinuationOfRiemannZetaUsingIntegral

Related topic

IntegralMeanValueTheorem

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example of conformal mapping

example of conformal mapping

Consider the four curves

A

=

{

t

}

,

B

=

{

t

+

i

⁢

t

}

,

C

=

{

i

⁢

t

}

and

D

=

{

-

t

+

i

⁢

t

}

,

t

∈

[

-

10

,

10

]

. Suppose there is a mapping

f

:

ℂ

↦

ℂ

which maps

A

to

D

and

B

to

C

. Is

f

conformal

at

z

0

=

0

? The size of the angles between

A

and

B

at the point of

intersection

z

0

=

0

is preserved, however the

orientation

is not. Therefore

f

is not conformal at

z

0

=

0

. Now suppose there is a

function

g

:

ℂ

↦

ℂ

which maps

A

to

C

and

B

to

D

. In this case we see not only that the size of the angles is preserved, but also the orientation. Therefore

g

is conformal at

z

0

=

0

.

Title

example of

conformal mapping

Canonical name

ExampleOfConformalMapping

Date of creation

2013-03-22 13:36:36

Last modified on

2013-03-22 13:36:36

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

6

Author

Johan (1032)

Entry type

Example

Classification

msc 30E20

Related topic

CategoryOfRiemannianManifolds

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

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example of using residue theorem

example of using residue theorem

We take an example of applying the

Cauchy residue theorem

in evaluating usual real improper integrals.

We shall the

integral

I

:=

∫

0

∞

cos

⁡

k

⁢

x

1

+

x

2

⁢

𝑑

x

where

k

is any

real number

.  One may prove that the integrand has no

antiderivative

among the

elementary functions

if

k

≠

0

.

Since the integrand is an even (

http://planetmath.org/EvenoddFunction

) and

x

↦

sin

⁡

k

⁢

x

1

+

x

2

an

odd function

, we may write

I

=

1

2

⁢

∫

-

∞

∞

cos

⁡

k

⁢

x

+

i

⁢

sin

⁡

k

⁢

x

1

+

x

2

⁢

𝑑

x

=

1

2

⁢

∫

-

∞

∞

e

i

⁢

k

⁢

x

1

+

x

2

⁢

𝑑

x

,

using also Euler’s formula.  Let’s consider the

contour integral

J

:=

∮

γ

e

i

⁢

k

⁢

z

1

+

z

2

⁢

𝑑

z

where

γ

is the perimeter of the semicircle consisting of the line segment from

(

-

R

,

0

)

to

(

R

,

0

)

and the semi-circular arc

c

connecting these points in the upper half-plane (

R

>

1

).  The integrand is

analytic

on and inside of

γ

except in the point

z

=

i

which is a

simple pole

.  Because we have (cf. the

coefficients of Laurent series

)

Res

⁡

(

e

i

⁢

k

⁢

z

1

+

z

2

,

i

)

=

lim

z

→

i

⁡

(

z

-

i

)

⁢

e

i

⁢

k

⁢

z

z

2

+

1

=

lim

z

→

i

⁡

e

i

⁢

k

⁢

z

z

+

i

=

e

-

k

2

⁢

i

,

the

residue theorem

(

http://planetmath.org/CauchyResidueTheorem

) yields

J

=

2

⁢

π

⁢

i

⋅

e

-

k

2

⁢

i

=

π

⁢

e

-

k

.

This does not depend on the radius

R

of the circle.

We split the integral

J

in two portions:  one along the

diameter

and the other along the circular arc

c

.  So we obtain

∫

-

R

R

e

i

⁢

k

⁢

x

1

+

x

2

⁢

𝑑

x

+

∫

c

e

i

⁢

k

⁢

z

1

+

z

2

⁢

𝑑

z

=

π

⁢

e

-

k

.

When

R

→

∞

,  the former portion tends to the limit

2

⁢

I

and the latter — as we at once shall see — to the limit 0.  Hence we get the result

I

=

∫

0

∞

cos

⁡

k

⁢

x

1

+

x

2

⁢

𝑑

x

=

π

2

⁢

e

k

.

As for the latter part of

J

, we denote

z

:=

x

+

i

⁢

y

(

x

,

y

∈

ℝ

);  then on the arc

c

, where

|

z

|

=

R

and

y

≧

0

, we have

|

e

i

⁢

k

⁢

z

1

+

z

2

|

=

|

e

-

k

⁢

y

+

i

⁢

k

⁢

x

|

|

1

+

z

2

|

=

e

-

k

⁢

y

|

1

+

z

2

|

≦

1

R

2

-

1

.

Using this estimation of the integrand we get, according the

integral estimating theorem

, the

inequality

|

∫

c

e

i

⁢

k

⁢

z

1

+

z

2

⁢

𝑑

z

|

≦

1

R

2

-

1

⋅

π

⁢

R

=

π

R

-

1

R

.

Since the right hand member tends to 0 as

R

→

∞

, then also the left hand member.

Title

example of using residue theorem

Canonical name

ExampleOfUsingResidueTheorem

Date of creation

2013-03-22 15:19:30

Last modified on

2013-03-22 15:19:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Example

Classification

msc 30E20

Related topic

ImproperIntegral

Related topic

Residue

Related topic

IntegralsOfEvenAndOddFunctions

Related topic

UsingResidueTheoremNearBranchPoint

Generated on Fri Feb 9 20:20:33 2018 by

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examples of infinite products

examples of infinite products

A classic example is the

Riemann zeta function

.
For

ℜ

⁡

(

z

)

>

1

we have

ζ

⁢

(

z

)

=

∑

n

=

1

∞

1

n

z

=

∏

p

⁢

prime

1

1

-

p

-

z

.

With the help of a

Fourier series

, or in other ways, one can prove
this infinite product expansion of the

sine function

:

sin

⁡

z

=

z

⁢

∏

n

=

1

∞

(

1

-

z

2

n

2

⁢

π

2

)

(1)

where

z

is an arbitrary

complex number

.
Taking the

logarithmic derivative

(a frequent move in connection with
infinite products) we get a decomposition
of the

cotangent

into

partial fractions

:

π

⁢

cot

⁡

π

⁢

z

=

1

z

+

∑

n

=

1

∞

(

1

z

+

n

+

1

z

-

n

)

.

(2)

The equation (

2

), in turn, has some interesting uses, e.g. to get
the

Taylor expansion

of an

Eisenstein series

, or to evaluate

ζ

⁢

(

2

⁢

n

)

for

positive integers

n

.

Title

examples of infinite products

Canonical name

ExamplesOfInfiniteProducts

Date of creation

2013-03-22 14:02:32

Last modified on

2013-03-22 14:02:32

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Example

Classification

msc 30E20

Related topic

ComplexTangentAndCotangent

Generated on Fri Feb 9 20:20:35 2018 by

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Gauss’ mean value theorem

Gauss’ mean value theorem

Let

Ω

be a domain in

ℂ

and suppose

f

is an

analytic function

on

Ω

.
Furthermore, let

C

be a circle inside

Ω

with center

z

0

and radius

r

. Then

f

⁢

(

z

0

)

is the mean value of

f

along

C

, that is,

f

⁢

(

z

0

)

=

1

2

⁢

π

⁢

∫

0

2

⁢

π

f

⁢

(

z

0

+

r

⁢

e

i

⁢

θ

)

⁢

𝑑

θ

.

Title

Gauss’ mean value theorem

Canonical name

GaussMeanValueTheorem

Date of creation

2013-03-22 13:35:33

Last modified on

2013-03-22 13:35:33

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

12

Author

Johan (1032)

Entry type

Theorem

Classification

msc 30E20

Related topic

GaussMeanValueTheoremForHarmonicFunctions

Related topic

AverageValueOfFunction

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

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