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The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behaviour with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems). In the abstract theory this is shown to come from a line bundle condition of descent.
The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half plane, which means it has positive imaginary part. It is given by the formula
It is convenient to define three auxiliary theta functions, which we may write
If we set z=0 in the above theta functions, we obtain four functions of τ only, defined on the upper half plane (sometimes called theta constants.) These can be used to define a variety of modular forms, and to parameterize certain curves; in particular the Jacobi identity is
The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since