The best-known properties and formulas for the Jacobi theta functions

Values for real arguments

For real values of the arguments , (with ), the values of the Jacobi theta functions , , , and are real.

For real values of the arguments , (with ), the values of the Jacobi theta functions , , , and are real.

Simple values at zero

All Jacobi theta functions , , , , , , , and have the following simple values at the origin point:

Specific values for specialized parameter

All Jacobi theta functions , , , , , , , and have the following simple values if :

At the points and , all theta functions , , , , , , , and can be expressed through the Dedekind eta function or a composition of the complete elliptic function and the inverse elliptic nome by the following formulas:

The previous relations can be generalized for the cases and , where :

Analyticity

All Jacobi theta functions , , , , , , , and are analytic functions of and for and .

Poles and essential singularities

All Jacobi theta functions , , , , , , , and do not have poles and essential singularities inside of the unit circle .

Branch points and branch cuts

For fixed , the functions , , , and have one branch point: . (The point is the branch cut endpoint.)

For fixed , the functions , , , and are the single‐valued functions inside the unit circle of the complex ‐plane, cut along the interval , where they are continuous from above:

For fixed , the functions , , , and do not have branch points and branch cuts with respect to .

The functions , , , and do not have branch points and branch cuts.

Natural boundary of analyticity

The unit circle is the natural boundary of the region of analyticity for all Jacobi theta functions , , , , , , , and .

Periodicity

The Jacobi theta functions and are the periodic functions with respect to with period and a quasi‐period :

The Jacobi theta functions and are the periodic functions with respect to with period and a quasi‐period :

The Jacobi theta functions and are the periodic functions with respect to with period :

The Jacobi theta functions and are the periodic functions with respect to with period :

The previous formulas are the particular cases of the following general relations that reflect the periodicity and quasi‐periodicity of the theta functions by variable :

Parity and symmetry

All Jacobi theta functions , , , , , , , and have mirror symmetry:

The Jacobi theta functions , , , and are odd functions with respect to :

The other Jacobi theta functions , , , and are even functions with respect to :

The Jacobi theta functions , , , and satisfy the following parity type relations with respect to :

The Jacobi theta functions , , , and with argument can be self-transformed by the following relations:

q-series representations

All Jacobi elliptic theta functions , , , and , and their derivatives , , , and have the following series expansions, which can be called ‐series representations:

Other series representations

The theta functions , , , and , and their derivatives , , , and can also be represented through the following series:

Product representations

The theta functions can be represented through infinite products, for example:

Transformations

The theta functions , , , and satisfy numerous relations that can provide transformations of their arguments, for example:

Among those transformations, several kinds can be combined into specially named groups:

root of :

Multiple angle formulas:

Double-angle formulas (which are not particular cases of the previous group), for example:

Landen's transformation:

Identities

The theta functions at satisfy numerous modular identities of the form , where the are positive integers and is a multivariate polynomials over the integers, for example:

Among the numerous identities for theta functions, several kinds can be joined into specially named groups:

Relations involving squares:

Relations involving quartic powers:

Relations between the four theta functions where the first argument is zero, for example:

Addition formulas:

Triple addition formulas, for example:

Representations of derivatives

The derivatives of the Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

The derivatives of the Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

The -order derivatives of the Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

The -order derivatives of Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

Integration

The indefinite integrals of the Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

The first four sums cannot be expressed in closed form through the named functions.

The indefinite integrals of the Jacobi theta functions , , , and , and their derivatives , , , and with respect to variable can be expressed by the following formulas:

Partial differential equations

The elliptic theta functions , , , and , and their derivatives , , , and satisfy the one‐dimensional heat equations:

The elliptic theta functions , , , and , and their derivatives , , , and satisfy the following second-order partial differential equations:

Zeros

The Jacobi theta functions , , , and , and their derivatives , , , and are equal to zero in the following points: