Jacobi theta functions

In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.

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 behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this comes from a line bundle condition of descent.

There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) 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

where $q = exp(π iτ)$ and $η = exp(2π iz)$ . It is a Jacobi form. If $τ$ is fixed, this becomes a Fourier series for a periodic entire function of $z$ with period 1; in this case, the theta function satisfies the identity

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