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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:
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