Theta function

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.[1]

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

where q = exp(π) is the nome and η = exp(2πiz). It is a Jacobi form. At fixed τ, this is a Fourier series for a 1-periodic entire function of z. Accordingly, the theta function is 1-periodic in z:

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:

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:

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 ττ + 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 (nn2 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 = eπ. 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.

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have

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 = eπ (noting some authors instead set q = e) and take w = eπiz then

We therefore obtain a product formula for the theta function in the form

where (  ;  ) is the q-Pochhammer symbol and θ(  ;  ) is the q-theta function. Expanding terms out, the Jacobi triple product can also be written

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:

was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform

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 fourth theta function – and thus the others too – is intimately connected to the Jackson q-gamma function via the relation[5]

Let η(τ) be the Dedekind eta function, and the argument of the theta function as the nome q = eπ. Then,

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.[6] Taking z = x to be real and τ = it with t real and positive, we can write

This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic delta function, or Dirac comb, in the sense of distributions

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

The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.

Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3.

This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.