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 the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions (notational variations) for further discussion.
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 parametrize certain curves; in particular, the Jacobi identity is
Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n ≡ n2 mod 2). For the second, let
Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q, where w = eπiz and q = eπiτ. In this form, the functions become
We see that the theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers.
It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.
If we express the theta function in terms of the nome q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then
We therefore obtain a product formula for the theta function in the form
This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
The Jacobi theta functions have the following integral representations:
These relations hold for all 0 < q < 1. Specializing the values of q, we have the next parameter free sums
All zeros of the Jacobi theta functions are simple zeros and are given by the following:
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 is given in the article on the Hurwitz zeta function.
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
where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of ℘(z) at z = 0 has zero constant term.
The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions. Taking z = x to be real and τ = it with t real and positive, we can write
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 with the theta function.
The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. This invariance is presented in the article on the theta representation of the Heisenberg group.
If F is a quadratic form in n variables, then the theta function associated with F is