aarjaneiro/planetmath huggingface dataset:
special cases of hypergeometric function

special cases of hypergeometric function

Many elementary (

http://planetmath.org/ElementaryFunction

) and non-elementary

transcendental functions

may be expressed as special cases of the

hypergeometric functions

F

⁢

(

a

,

b

,

c

;

x

)

=

1

+

a

⁢

b

1

!

⁢

c

⁢

x

+

a

⁢

(

a

+

1

)

⁢

b

⁢

(

b

+

1

)

2

!

⁢

c

⁢

(

c

+

1

)

⁢

x

2

+

a

⁢

(

a

+

1

)

⁢

(

a

+

2

)

⁢

b

⁢

(

b

+

1

)

⁢

(

b

+

2

)

3

!

⁢

c

⁢

(

c

+

1

)

⁢

(

c

+

2

)

⁢

x

3

+

…

,

which are solutions of the

hypergeometric equation

x

⁢

(

x

-

1

)

⁢

d

2

⁢

y

d

⁢

x

2

+

(

c

-

(

a

+

b

+

1

)

)

⁢

d

⁢

y

d

⁢

x

-

a

⁢

b

⁢

y

=

0

.

For example:

•

(

1

+

x

)

n

=

F

⁢

(

-

n

,

1

,

1

;

-

x

)

•

ln

⁡

(

1

+

x

)

=

x

⁢

F

⁢

(

1

,

1

,

2

;

-

x

)

•

ln

⁡

1

+

x

1

-

x

=

2

⁢

x

⁢

F

⁢

(

1

2

,

1

,

3

2

;

x

2

)

•

arcsin

⁡

x

=

x

⁢

F

⁢

(

1

2

,

1

2

,

3

2

;

x

2

)

•

arctan

⁡

x

=

x

⁢

F

⁢

(

1

2

,

1

,

3

2

;

-

x

2

)

•

sin

⁡

(

m

⁢

arcsin

⁡

x

)

=

m

⁢

x

⁢

F

⁢

(

1

+

m

2

,

1

-

m

2

,

3

2

;

x

2

)

•

cos

⁡

(

m

⁢

arcsin

⁡

x

)

=

F

⁢

(

m

2

,

-

m

2

,

1

2

;

x

2

)

•

T

n

⁢

(

x

)

=

F

⁢

(

n

,

-

n

,

1

2

;

1

-

x

2

)

(

Chebyshev polynomials

)

•

P

n

⁢

(

x

)

=

F

⁢

(

-

n

,

n

+

1

,

1

;

1

-

x

2

)

(

Legendre polynomials

)

•

∫

0

π

2

d

⁢

φ

1

-

x

2

⁢

sin

2

⁡

φ

=

π

2

⁢

F

⁢

(

1

2

,

1

2

,

1

;

x

2

)

(

complete elliptic integral

of 1st kind)

•

∫

0

π

2

1

-

x

2

⁢

sin

2

⁡

φ

⁢

𝑑

φ

=

π

2

⁢

F

⁢

(

-

1

2

,

1

2

,

1

;

x

2

)

(complete elliptic integral of 2nd kind)

Title

special cases of hypergeometric function

Canonical name

SpecialCasesOfHypergeometricFunction

Date of creation

2013-03-22 18:54:39

Last modified on

2013-03-22 18:54:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 33C05

Related topic

FrobeniusMethod

Related topic

IndexOfSpecialFunctions

Related topic

GettingTaylorSeriesFromDifferentialEquation

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

LaTeXML
Legendre polynomial

Legendre polynomial

The

Legendre polynomials

are a set of polynomials

{

P

n

}

n

=

0

∞

each of order

n

that satisfy Legendre’s ODE:

d

d

⁢

x

⁢

[

(

1

-

x

2

)

⁢

P

n

′

⁢

(

x

)

]

+

n

⁢

(

n

+

1

)

⁢

P

n

⁢

(

x

)

=

0

.

Alternatively

P

n

is an

eigenfunction

of the

self-adjoint

differential operator

d

d

⁢

x

⁢

(

1

-

x

2

)

⁢

d

d

⁢

x

with

eigenvalue

-

n

⁢

(

n

+

1

)

.

The Legendre polynomials are also known as Legendre functions of the

first kind

.

By Sturm-Liouville theory, this means they’re

orthogonal

over some interval with
some weight function. In fact it can be shown that they’re orthogonal on

[

-

1

,

1

]

with weight function

W

⁢

(

x

)

=

1

. As with any set of

orthogonal polynomials

, this can be used to generate them (up to normalization) by

Gram-Schmidt orthogonalization

of the monomials

{

x

i

}

. The normalization used
is

⟨

P

n

∥

P

n

⟩

=

2

/

(

2

⁢

n

+

1

)

, which makes

P

n

⁢

(

±

1

)

=

(

±

1

)

n

Rodrigues’s Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of

P

n

in terms of

derivatives

:

P

n

⁢

(

x

)

=

1

2

n

⁢

n

!

⁢

(

d

d

⁢

x

)

n

⁢

(

x

2

-

1

)

n

The first few explicitly are:

P

0

⁢

(

x

)

=

1

P

1

⁢

(

x

)

=

x

P

2

⁢

(

x

)

=

1

2

⁢

(

3

⁢

x

2

-

1

)

P

3

⁢

(

x

)

=

1

2

⁢

(

5

⁢

x

3

-

3

⁢

x

)

P

4

⁢

(

x

)

=

1

8

⁢

(

35

⁢

x

4

-

30

⁢

x

2

+

3

)

…

As all orthogonal polynomials do, these satisfy a three-term

recurrence relation

:

(

n

+

1

)

⁢

P

n

+

1

⁢

(

x

)

=

(

2

⁢

n

+

1

)

⁢

x

⁢

P

n

⁢

(

x

)

-

(

n

)

⁢

P

n

-

1

⁢

(

x

)

The Legendre functions of the

second kind

also satisfy the Legendre ODE but are not regular at the origin.

Related are the associated Legendre functions, and

spherical harmonics

.

Title

Legendre polynomial

Canonical name

LegendrePolynomial

Date of creation

2013-03-22 15:12:10

Last modified on

2013-03-22 15:12:10

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 33C45

Related topic

OrthogonalPolynomials

Defines

Rodrigues’s Formula

Defines

Legendre’s

Differential Equation

Generated on Fri Feb 9 20:36:49 2018 by

LaTeXML
orthogonality of Chebyshev polynomials from recursion

orthogonality of Chebyshev polynomials from recursion

In this entry, we shall demonstrate the orthogonality
relation of the

Chebyshev polynomials

from their

recursion

relation

. Recall that this relation reads as

T

n

+

1

⁢

(

x

)

-

2

⁢

x

⁢

T

n

⁢

(

x

)

+

T

n

-

1

=

0

with

initial conditions

T

0

⁢

(

x

)

=

1

and

T

1

⁢

(

x

)

=

x

.
The relation we seek to demonstrate is

∫

-

1

+

1

𝑑

x

⁢

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

1

-

x

2

=

0

when

m

≠

n

.

We start with the observation that

T

n

is an

even function

when

n

is even and an

odd function

when

n

is odd. That
this is true for

T

0

and

T

1

follows immediately from their
definitions. When

n

>

1

, we may induce this from the
recursion. Suppose that

T

m

⁢

(

-

x

)

=

(

-

1

)

m

⁢

T

m

⁢

(

x

)

when

m

<

n

. Then we have

T

n

+

1

⁢

(

-

x

)

=

2

⁢

(

-

x

)

⁢

T

n

⁢

(

-

x

)

-

T

n

-

1

⁢

(

-

x

)

=

-

(

-

1

)

n

⁢

2

⁢

x

⁢

T

n

⁢

(

x

)

-

(

-

1

)

n

-

1

⁢

T

n

-

1

⁢

(

x

)

=

(

-

1

)

n

+

1

⁢

(

2

⁢

x

⁢

T

n

⁢

(

x

)

-

T

n

-

1

⁢

(

x

)

)

=

(

-

1

)

n

+

1

⁢

T

n

+

1

⁢

(

x

)

.

From this observation, we may immediately conclude half
of orthogonality. Suppose that

m

and

n

are nonnegative
integers whose

difference

is odd. Then

T

m

⁢

(

-

x

)

⁢

T

n

⁢

(

-

x

)

=

-

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

, so we have

∫

-

1

+

1

𝑑

x

⁢

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

1

-

x

2

=

0

because the integrand is an odd function of

x

.

To cover the remaining cases, we shall proceed by

induction

.
Assume that

T

k

is

orthogonal

to

T

m

whenever

m

≤

n

and

k

≤

n

and

m

≠

k

. By the

conclusions

of last
paragraph, we know that

T

n

+

1

is orthogonal to

T

n

.
Assume then that

m

≤

n

-

1

. Using the recursion, we have

∫

-

1

+

1

𝑑

x

⁢

T

m

⁢

(

x

)

⁢

T

n

+

1

⁢

(

x

)

1

-

x

2

=

2

⁢

∫

-

1

+

1

𝑑

x

⁢

x

⁢

T

m

⁢

(

x

)

⁢

T

n

⁢

(

x

)

1

-

x

2

-

∫

-

1

+

1

𝑑

x

⁢

T

m

⁢

(

x

)

⁢

T

n

-

1

⁢

(

x

)

1

-

x

2

=

∫

-

1

+

1

𝑑

x

⁢

T

m

+

1

⁢

(

x

)

⁢

T

n

⁢

(

x

)

1

-

x

2

+

∫

-

1

+

1

𝑑

x

⁢

T

m

-

1

⁢

(

x

)

⁢

T

n

⁢

(

x

)

1

-

x

2

-

∫

-

1

+

1

𝑑

x

⁢

T

m

⁢

(

x

)

⁢

T

n

-

1

⁢

(

x

)

1

-

x

2

By our

assumption

, each of the three

integrals

is zero,
hence

T

n

+

1

is orthogonal to

T

m

, so we conclude
that

T

k

is orthogonal to

T

m

when

m

≤

n

+

1

and

k

≤

n

+

1

and

m

≠

k

.

Title

orthogonality of Chebyshev polynomials from recursion

Canonical name

OrthogonalityOfChebyshevPolynomialsFromRecursion

Date of creation

2013-03-22 18:54:46

Last modified on

2013-03-22 18:54:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 33C45

Classification

msc 33D45

Classification

msc 42C05

Generated on Fri Feb 9 20:36:58 2018 by

LaTeXML
orthogonality of Laguerre polynomials

orthogonality of Laguerre polynomials

We use the

definition

of

Laguerre polynomials

L

n

⁢

(

x

)

via their

Rodrigues formula

(

http://planetmath.org/RodriguesFormula

)

L

n

⁢

(

x

)

:=

e

x

⁢

d

n

d

⁢

x

n

⁢

(

x

n

⁢

e

-

x

)

.

(1)

The

polynomials

(1) themselves are not

orthogonal

to each other, but the

expressions

e

-

x

2

⁢

L

n

⁢

(

x

)

(

n

=

0

,

1

,

2

,

…

) are orthogonal (

http://planetmath.org/OrthogonalPolynomials

) on the

interval

from 0 to

∞

, i.e. the polynomials are orthogonal with respect to the

weighting function

e

-

x

on that interval, as is seen in the following.

Let

m

be another

nonnegative integer

.  We integrate by parts (

http://planetmath.org/IntegrationByParts

)

m

times in

∫

0

∞

e

-

x

⁢

x

m

⁢

L

n

⁢

(

x

)

⁢

𝑑

x

=

∫

0

∞

x

m

⁢

d

n

d

⁢

x

n

⁢

(

x

n

⁢

e

-

x

)

⁢

𝑑

x

=

(

-

1

)

m

⁢

m

!

⁢

∫

0

∞

d

n

-

m

d

⁢

x

n

-

m

⁢

(

x

m

⁢

e

-

x

)

⁢

𝑑

x

.

When

m

<

n

,  this yields

∫

0

∞

e

-

x

⁢

x

m

⁢

L

n

⁢

(

x

)

⁢

𝑑

x

=

(

-

1

)

m

⁢

m

!

⁢

/

x

=

0

∞

⁡

d

n

-

m

-

1

d

⁢

x

n

-

m

-

1

⁢

(

x

m

⁢

e

-

x

)

=

0

.

(2)

and for

m

=

n

it gives

∫

0

∞

e

-

x

⁢

x

m

⁢

L

n

⁢

(

x

)

⁢

𝑑

x

=

(

-

1

)

n

⁢

n

!

⁢

∫

0

∞

x

n

⁢

e

-

x

⁢

𝑑

x

=

(

-

1

)

n

⁢

(

n

!

)

2

.

(3)

The result (2) implies, because

L

m

⁢

(

x

)

is a polynomial of degree

m

, that

∫

0

∞

e

-

x

L

m

(

x

)

L

n

(

x

)

d

x

=

0

(

m

<

n

)

,

whence also

∫

0

∞

e

-

x

L

m

(

x

)

L

n

(

x

)

d

x

=

0

(

m

≠

n

)

.

(4)

Thus the orthogonality has been shown.  Therefore, since the leading term of

L

n

⁢

(

x

)

is

(

-

1

)

n

⁢

x

n

, we infer by (3) and (4) that

∫

0

∞

e

-

x

⁢

[

L

n

⁢

(

x

)

]

2

⁢

𝑑

x

=

(

-

1

)

n

⁢

∫

0

∞

e

-

x

⁢

x

n

⁢

L

n

⁢

(

x

)

⁢

𝑑

s

=

(

n

!

)

2

,

so that the expressions

L

n

⁢

(

x

)

n

!

form a system of orthonormal polynomials.

References

1

H. Eyring, J. Walter, G. Kimball:

Quantum chemistry

.  Eight printing.  Wiley & Sons, New York (1958).

Title

orthogonality of Laguerre polynomials

Canonical

name

OrthogonalityOfLaguerrePolynomials

Date of creation

2013-03-22 19:05:50

Last modified on

2013-03-22 19:05:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Derivation

Classification

msc 33D45

Classification

msc 33C45

Classification

msc 26C05

Related topic

SubstitutionNotation

Related topic

PropertiesOfOrthogonalPolynomials

Generated on Fri Feb 9 20:37:09 2018 by

LaTeXML
convergence of arithmetic-geometric mean

convergence of arithmetic-geometric mean

In this entry, we show that the

arithmetic-geometric mean

converges

.
By the

arithmetic-geometric means inequality

, we know that the

sequences

of

arithmetic

and

geometric means

are both

monotonic

and

bounded

, so
they converge individually. What still needs to be shown is that they
converge to the same limit.

Define

x

n

=

a

n

/

g

n

. By the arithmetic-geometric

inequality

, we
have

x

n

≥

1

. By the defining

recursions

, we have

x

n

+

1

=

a

n

+

1

g

n

+

1

=

a

n

+

g

n

2

⁢

a

n

⁢

g

n

=

1

2

⁢

(

a

n

g

n

+

g

n

a

n

)

=

1

2

⁢

(

x

n

+

1

x

n

)

Since

x

n

≥

1

, we have

1

/

x

n

≤

1

, and

x

n

≤

x

n

, hence

x

n

+

1

-

1

=

1

2

⁢

(

x

n

+

1

x

n

-

2

)

≤

1

2

⁢

(

x

n

+

1

-

2

)

≤

1

2

⁢

(

x

n

-

1

)

.

From this inequality

0

≤

x

n

+

1

-

1

≤

1

2

⁢

(

x

n

-

1

)

,

we may conclude that

x

n

→

1

as

n

→

∞

, which , by the

definition

of

x

n

,
is

equivalent

to

lim

n

→

∞

⁡

g

n

=

lim

n

→

∞

⁡

a

n

.

Not only have we proven that the arithmetic-geometric mean converges, but we can infer
a rate of convergence from our proof. Namely, we have that

0

≤

x

n

-

1

≤

(

x

0

-

1

)

/

2

n

. Hence, we see that the rate of convergence of

a

n

and

g

n

to the answer goes
as

O

⁢

(

2

-

n

)

.

By more carefully bounding the recursion for

x

n

above, we may obtain better estimates
of the rate of convergence. We will now derive an inequality. Suppose that

y

≥

0

.

0

≤

y

5

+

y

4

+

4

⁢

y

3

+

3

⁢

y

2

y

2

+

4

⁢

y

+

4

≤

y

5

+

y

4

+

4

⁢

y

3

+

4

⁢

y

2

+

4

⁢

y

+

4

(

y

+

2

)

2

≤

(

y

+

1

)

⁢

(

y

2

+

2

)

2

Set

x

=

y

+

1

(so we have

x

≥

1

).

(

x

+

1

)

2

≤

x

⁢

(

(

x

-

1

)

2

+

2

)

2

x

≤

x

2

⁢

(

(

x

-

1

)

2

+

2

)

2

(

x

+

1

)

2

x

≤

x

⁢

(

(

x

-

1

)

2

+

2

)

x

+

1

x

+

1

x

⁢

x

≤

(

x

-

1

)

2

+

2

1

2

⁢

(

x

+

1

x

)

≤

1

+

1

2

⁢

(

x

-

1

)

2

Thus, because

x

n

+

1

=

(

x

n

+

1

/

x

n

)

/

2

, we have

x

n

+

1

-

1

≤

1

2

⁢

(

x

n

-

1

)

2

.

From this

equation

, we may derive the bound

x

n

-

1

≤

1

2

2

n

-

1

⁢

(

x

0

-

1

)

2

n

.

This is a much better bound! It approaches zero far more rapidly
than any

exponential function

, so we have superlinear convergence.

Title

convergence of arithmetic-geometric mean

Canonical

name

ConvergenceOfArithmeticgeometricMean

Date of creation

2013-03-22 17:09:46

Last modified on

2013-03-22 17:09:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Classification

msc 26E60

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

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elliptic function

elliptic function

Let

Λ

∈

ℂ

be a

lattice

in the sense of

number
theory

, i.e. a 2-dimensional

free group

over

ℤ

which
generates

ℂ

over

ℝ

.

An

elliptic function

ϕ

, with respect to the lattice

Λ

, is a

meromorphic

funtion

ϕ

:

ℂ

→

ℂ

which is

Λ

-

periodic

:

ϕ

⁢

(

z

+

λ

)

=

ϕ

⁢

(

z

)

,

∀

z

∈

ℂ

,

∀

λ

∈

Λ

Remark

: An elliptic function which is

holomorphic

is
constant. Indeed such a

function

would induce a

holomorphic
function

on

ℂ

/

Λ

, which is

compact

(and it is a
standard result from

Complex Analysis

that any holomorphic
function with compact domain is constant, this follows from
Liouville’s Theorem).

Example

: The Weierstrass

℘

-function (see

elliptic curve

)
is an elliptic function, probably the most important. In fact:

Theorem 1

.

The field of elliptic functions with respect to a lattice

Λ

is generated by

℘

and

℘

′

(the derivative of

℘

).

Proof.

See

[

2

]

, chapter 1, theorem 4.
∎

References

1

James Milne,

Modular Functions

and

Modular Forms

, online course notes.

http://www.jmilne.org/math/CourseNotes/math678.html

http://www.jmilne.org/math/CourseNotes/math678.html

2

Serge Lang,

Elliptic Functions

. Springer-Verlag, New York.

3

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

Title

elliptic function

Canonical name

EllipticFunction

Date of creation

2013-03-22 13:47:03

Last modified on

2013-03-22 13:47:03

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Definition

Classification

msc 33E05

Related topic

ExamplesOfEllipticFunctions

Related topic

WeierstrassSigmaFunction

Related topic

ModularDiscriminant

Related topic

WeierstrassWpFunction

Related topic

TableOfMittagLefflerPartialFractionExpansions

Related topic

PeriodicFunctions

Generated on Fri Feb 9 20:37:24 2018 by

LaTeXML
examples of elliptic functions

examples of elliptic functions

Examples of Elliptic Functions

Let

Λ

⊂

ℂ

be a

lattice

generated by

w

1

,

w

2

. Let

Λ

∗

denote

Λ

-

{

0

}

.

1.

The

Weierstrass

℘

-

function

is defined by the
series

℘

⁢

(

z

;

Λ

)

=

1

z

2

+

∑

w

∈

Λ

∗

1

(

z

-

w

)

2

-

1

w

2

2.

The

derivative

of the Weierstrass

℘

-function is also an

elliptic function

℘

′

⁢

(

z

;

Λ

)

=

-

2

⁢

∑

w

∈

Λ

∗

1

(

z

-

w

)

3

3.

The

Eisenstein series of weight

2

⁢

k

for

Λ

is
the series

𝒢

2

⁢

k

⁢

(

Λ

)

=

∑

w

∈

Λ

∗

w

-

2

⁢

k

The Eisenstein series of weight

4

and

6

are of special
relevance in the theory of elliptic curves. In particular, the quantities

g

2

and

g

3

are usually defined as follows:

g

2

=

60

⋅

𝒢

4

⁢

(

Λ

)

,

g

3

=

140

⋅

𝒢

6

⁢

(

Λ

)

Remark:

The elliptic functions

℘

,

℘

′

and

𝒢

2

⁢

k

are related by the following important

equation

:

(

℘

′

⁢

(

z

;

Λ

)

)

2

=

4

⁢

℘

⁢

(

z

;

Λ

)

3

-

g

2

⁢

(

Λ

)

⁢

℘

⁢

(

z

;

Λ

)

-

g

3

⁢

(

Λ

)

In particular, the previous equation provides an

isomorphism

between

ℂ

/

Λ

and the elliptic curve

E

:

y

2

=

4

⁢

x

3

-

g

2

⁢

x

-

g

3

given by:

ℂ

/

Λ

→

E

,

z

↦

(

℘

⁢

(

z

;

Λ

)

,

℘

′

⁢

(

z

;

Λ

)

)

.

Title

examples of elliptic functions

Canonical name

ExamplesOfEllipticFunctions

Date of creation

2013-03-22 13:54:04

Last modified on

2013-03-22 13:54:04

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Example

Classification

msc 33E05

Related topic

EllipticFunction

Related topic

WeierstrassWpFunction

Defines

Eisenstein series

Generated on Fri Feb 9 20:37:29 2018 by

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integral represetations of Jacobi ϑ functions

integral represetations of Jacobi

ϑ

functions

The

Jacobi theta functions

have the following

integral

representations:

ϑ

1

(

z

|

τ

)

=

-

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

∫

i

-

∞

i

+

∞

e

i

⁢

π

⁢

τ

⁢

u

2

⁢

cos

⁡

(

2

⁢

u

⁢

z

+

π

⁢

τ

⁢

u

)

sin

⁡

(

π

⁢

u

)

d

u

ϑ

2

(

z

|

τ

)

=

-

i

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

∫

i

-

∞

i

+

∞

e

i

⁢

π

⁢

τ

⁢

u

2

⁢

cos

⁡

(

2

⁢

u

⁢

z

+

π

⁢

u

+

π

⁢

τ

⁢

u

)

sin

⁡

(

π

⁢

u

)

d

u

ϑ

3

(

z

|

τ

)

=

-

i

∫

i

-

∞

i

+

∞

e

i

⁢

π

⁢

τ

⁢

u

2

⁢

cos

⁡

(

2

⁢

u

⁢

z

+

π

⁢

u

)

sin

⁡

(

π

⁢

u

)

d

u

ϑ

4

(

z

|

τ

)

=

-

i

∫

i

-

∞

i

+

∞

e

i

⁢

π

⁢

τ

⁢

u

2

⁢

cos

⁡

(

2

⁢

u

⁢

z

)

sin

⁡

(

π

⁢

u

)

d

u

Title

integral represetations of Jacobi

ϑ

functions

Canonical name

IntegralRepresetationsOfJacobivarthetaFunctions

Date of creation

2013-03-22 14:39:52

Last modified on

2013-03-22 14:39:52

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Generated on Fri Feb 9 20:37:31 2018 by

LaTeXML
Jacobi’s identity for ϑ functions

Jacobi’s identity for

ϑ

functions

Jacobi’s

identities

describe how

theta functions

transform under replacing the period with the negative of its reciprocal. Together with the

quasiperiodicity

relations, they describe the transformations of theta functions under the

modular group

.

θ

1

(

z

∣

-

1

/

τ

)

=

-

i

(

-

i

τ

)

1

/

2

e

i

⁢

τ

⁢

z

2

π

θ

1

(

τ

z

∣

τ

)

θ

2

(

z

∣

-

1

/

τ

)

=

(

-

i

τ

)

1

/

2

e

i

⁢

τ

⁢

z

2

π

θ

4

(

τ

z

∣

τ

)

θ

3

(

z

∣

-

1

/

τ

)

=

(

-

i

τ

)

1

/

2

e

i

⁢

τ

⁢

z

2

π

θ

3

(

τ

z

∣

τ

)

θ

4

(

z

∣

-

1

/

τ

)

=

(

-

i

τ

)

1

/

2

e

i

⁢

τ

⁢

z

2

π

θ

2

(

τ

z

∣

τ

)

Title

Jacobi’s identity for

ϑ

functions

Canonical name

JacobisIdentityForvarthetaFunctions

Date of creation

2013-03-22 14:46:45

Last modified on

2013-03-22 14:46:45

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Generated on Fri Feb 9 20:37:33 2018 by

LaTeXML
modular discriminant

modular discriminant

Definition 1

.

Let

Λ

⊂

C

be a

lattice

.

1.

Let

q

τ

=

e

2

⁢

π

⁢

i

⁢

τ

. The

Dedekind eta
function

is defined to be

η

⁢

(

τ

)

=

q

τ

1

/

24

⁢

∏

n

=

1

∞

(

1

-

q

τ

n

)

The Dedekind eta function should not be confused with the

Weierstrass eta function

,

η

⁢

(

w

;

Λ

)

.

2.

The

j

-

invariant

, as a

function

of lattices, is defined to
be:

j

⁢

(

Λ

)

=

g

2

3

g

2

3

-

27

⁢

g

3

2

where

g

2

and

g

3

are certain

multiples

of the

Eisenstein
series

of weight

4

and

6

(see

http://planetmath.org/encyclopedia/ExamplesOfEllipticFunctions.html

this
entry).

3.

The

Δ

function

(

delta function

or

modular discriminant

) is defined to be

Δ

⁢

(

Λ

)

=

g

2

3

-

27

⁢

g

3

2

Let

Λ

τ

be the lattice generated by

1

,

τ

. The

Δ

function for

Λ

τ

has a product expansion

Δ

⁢

(

τ

)

=

Δ

⁢

(

Λ

τ

)

=

(

2

⁢

π

⁢

i

)

12

⁢

q

τ

⁢

∏

n

=

1

∞

(

1

-

q

τ

n

)

24

=

(

2

⁢

π

⁢

i

)

12

⁢

η

⁢

(

τ

)

24

Title

modular discriminant

Canonical name

ModularDiscriminant

Date of creation

2013-03-22 13:54:09

Last modified on

2013-03-22 13:54:09

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 33E05

Synonym

delta function

Related topic

EllipticFunction

Related topic

JInvariant

Related topic

WeierstrassSigmaFunction

Related topic

Discriminant

Related topic

DiscriminantOfANumberField

Related topic

RamanujanTauFunction

Defines

modular discriminant

Defines

Dedekind eta function

Generated on Fri Feb 9 20:37:39 2018 by

LaTeXML
order of an elliptic function

order of an elliptic function

The

order

of an

elliptic function

is the number of poles of the

function

contained

within a fundamental period parallelogram, counted with

multiplicity

.
Sometimes the term “degree” is also used — this usage agrees with the
theory of

Riemann surfaces

.

This order is always a finite number; this follows from the fact that a

meromorphic function

can only have a finite number of poles in a

compact

region (such as the

closure

of a period parallelogram). As it turns out, the
order can be any integer greater than 1.

Title

order of an elliptic function

Canonical name

OrderOfAnEllipticFunction

Date of creation

2013-03-22 15:44:35

Last modified on

2013-03-22 15:44:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33E05

Generated on Fri Feb 9 20:37:41 2018 by

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