What is "theta function"

Theta \The"ta\, n. [L., fr. Gr. qh^ta, the Greek letter [theta], A letter of the Greek alphabet corresponding to th in English; -- sometimes called the unlucky letter, from being used by the judges on their ballots in passing condemnation on a prisoner, it being the first letter of the Greek qa`natos, death.

Theta function (Math.), one of a group of functions used in developing the properties of elliptic functions.

n. (context math English) Any of a group of special functions of several complex variables, important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms.

400px|thumb|upright=1.2|Jacobi's original theta function θ with u = iπz and with nome q = e = 0.1e. Conventions are (Mathematica)θ(u; q) = 2q∑( − 1)qsin(2n + 1)u this is: θ(u; q) = ∑( − 1)qe 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.

Theta functions ϑ(z; τ) are special functions of several complex variables. Theta function may also refer to:

Jacobi-related functions:

• Q-theta function θ(z; q), a type of q-series
• Theta function of a lattice Θ(τ), a holomorphic function on the upper half-plane
• Mock theta functions μ(q), f(q), ϕ(q), ψ(q), χ(q), etc., a mock modular form of weight 1/2
• Ramanujan theta function f(a, b)
• Neville theta functions

Other functions:

• Riemann–Siegel theta function θ(t)
• Chebyshev function ϑ(x)
• Feferman's function θ(β)
• Heaviside step function sometimes denoted θ(x)
• Lovász theta function, an upper bound on the Shannon capacity of a graph