Jacobi ϑ functions

Jacobi

ϑ

functions

The Jacobi

ϑ

functions

are 4 basic functions of Jacobi’s theory of

elliptic functions

. They are functions of two complex variables,

z

and

τ

, known as the

argument

and the

half-period ratio

, respectively. It is often convenient to use the quantity

q

=

e

π

⁢

i

⁢

τ

, which is known as the

nome

. When the half-period ratio is used, these functions are denoted

ϑ

j

(

z

|

τ

)

(the index

j

runs from 1 to 4), and when the nome is used, the functions are denoted

ϑ

j

⁢

(

z

;

q

)

.

q

and

τ

are sometimes snipped for brevity when they are obvious from the context, and when that is done, the functions are denoted

ϑ

j

⁢

(

z

)

.

These functions can be defined by the following series. It is also possible to express them as products and as

integrals

(

http://planetmath.org/IntegralRepresetationsOfJacobiVarthetaFunctions

) — see the attachments to this entry for details.

ϑ

1

⁢

(

z

;

q

)

=

∑

n

=

-

∞

∞

(

-

1

)

n

⁢

q

(

n

+

1

/

2

)

2

⁢

e

(

2

⁢

n

+

1

)

⁢

i

⁢

z

=

2

⁢

∑

n

=

0

∞

(

-

1

)

n

⁢

q

(

n

+

1

/

2

)

2

⁢

sin

⁡

(

2

⁢

n

+

1

)

⁢

z

(1)

ϑ

1

(

z

∣

τ

)

=

∑

n

=

-

∞

∞

(

-

1

)

n

e

i

⁢

π

⁢

τ

⁢

(

n

+

1

/

2

)

2

+

(

2

⁢

n

+

1

)

⁢

i

⁢

z

(2)

ϑ

2

⁢

(

z

;

q

)

=

∑

n

=

-

∞

∞

q

(

n

+

1

/

2

)

2

⁢

e

(

2

⁢

n

+

1

)

⁢

i

⁢

z

=

2

⁢

∑

n

=

0

∞

q

(

n

+

1

/

2

)

2

⁢

cos

⁡

(

2

⁢

n

+

1

)

⁢

z

(3)

ϑ

2

(

z

∣

τ

)

=

∑

n

=

-

∞

∞

e

i

⁢

π

⁢

τ

⁢

(

n

+

1

/

2

)

2

+

(

2

⁢

n

+

1

)

⁢

i

⁢

z

(4)

ϑ

3

⁢

(

z

;

q

)

=

∑

n

=

-

∞

∞

q

n

2

⁢

e

2

⁢

i

⁢

n

⁢

z

=

1

+

2

⁢

∑

n

=

1

∞

q

n

2

⁢

cos

⁡

(

2

⁢

n

⁢

z

)

(5)

ϑ

3

(

z

∣

τ

)

=

∑

n

=

-

∞

∞

e

i

⁢

π

⁢

τ

⁢

n

2

+

2

⁢

i

⁢

n

⁢

z

(6)

ϑ

4

⁢

(

z

;

q

)

=

∑

n

=

-

∞

∞

(

-

1

)

n

⁢

q

n

2

⁢

e

2

⁢

i

⁢

n

⁢

z

=

1

+

2

⁢

∑

n

=

1

∞

(

-

1

)

n

⁢

q

n

2

⁢

cos

⁡

(

2

⁢

n

⁢

z

)

(7)

ϑ

4

(

z

∣

τ

)

=

∑

n

=

-

∞

∞

(

-

1

)

n

e

i

⁢

π

⁢

τ

⁢

n

2

+

2

⁢

i

⁢

n

⁢

z

(8)

Note that these series

converge

for all complex values of

z

whenever

|

q

|

<

1

(equivalently, when

ℑ

⁡

τ

>

0

). Furthermore, these series

converge uniformly

on compact subsets (this may be shown using the

Weierstrass M-test

) so these functions are

analytic

.

The

theta functions

satisfy many

identities

, the most important of which are the

quasiperiodicity

identities (

http://planetmath.org/QuasiperiodAndHalfquasipreiodRelationsForJacobiVarthetaFunctions

), Jacobi’s identity (

http://planetmath.org/JacobisIdentityForVarthetaFunctions

), and Landen’s

transformation

. For these identities and others, please see the attachments.

The theta functions were originally introduced because it is possible to express the

Jacobi elliptic functions

as ratios of theta functions. In some ways, this role is

similar

to the role the complex

exponential

plays in the theory of

trigonometric functions

. Just as one can derive complicated

trigonometric identities

form the properties of the

exponential functions

, so too one can derive complicated identites for elliptic functions using the properties of theta functions.

They are very useful in the numerical analysis of elliptic functions, since the series given above converge rapidly. Hence (as was realized early on by Jacobi), it is usually better to compute elliptic functions by first computing theta functions.

In

addition

, theta functions are interesting in their own right and appear in numerous, often surprising contexts, as the following exampes show. Theta functions appear as Green’s functions for the heat equation. In

number theory

, they are used to study the

representations

of

integers

as sums of squares. Theta functions can be used to construct

modular functions

. They can be used to construct integral representations of generating functions. In theoretical physics, they are used to perform sums over crystals and describe hexagonal lattices of vortices.

Title

Jacobi

ϑ

functions

Canonical name

JacobivarthetaFunctions

Date of creation

2013-03-22 14:08:10

Last modified on

2013-03-22 14:08:10

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

21

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33E05

Defines

Jacobi theta functions

Defines

nome

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