The bestknown 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 selftransformed by the following relations:
qseries 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:
Doubleangle 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 secondorder partial differential equations:
Zeros
The Jacobi theta functions , , , and , and their derivatives , , , and are equal to zero in the following points:
