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.
The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables, 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.

Jacobi theta function

There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them.
One Jacobi theta function is a function defined for two complex variables and, where can be any complex number and 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 is the nome and. It is a Jacobi form. At fixed, this is a Fourier series for a 1-periodic entire function of. Accordingly, the theta function is 1-periodic in :
It also turns out to be -quasiperiodic in, with
Thus, in general,
for any integers and.

Auxiliary functions

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 auxiliary functions are defined by
This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome rather than. In Jacobi's notation the -functions are written:
The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions for further discussion.
If we set in the above theta functions, we obtain four functions of only, defined on the upper half-plane These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
which is the Fermat curve of degree four.

Jacobi identities

Jacobi's identities describe how theta functions transform under the modular group, which is generated by and. Equations for the first transform are easily found since adding one to in the exponent has the same effect as adding to . For the second, let
Then

Theta functions in terms of the nome

Instead of expressing the Theta functions in terms of and, we may express them in terms of arguments and the nome, where and. In this form, the functions become
We see that the theta functions can also be defined in terms of and, 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 -adic numbers.

Product representations

The Jacobi triple product tells us that for complex numbers and with and 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 and take then
We therefore obtain a product formula for the theta function in the form
In terms of and :
where is the -Pochhammer symbol and is the -theta function. Expanding terms out, the Jacobi triple product can also be written
which we may also write as
This form is valid in general but clearly is of particular interest when is real. Similar product formulas for the auxiliary theta functions are

Integral representations

The Jacobi theta functions have the following integral representations:

Explicit values

See Yi.

Some series identities

The next two series identities were proved by István Mező:
These relations hold for all. Specializing the values of, we have the next parameter free sums

Zeros of the Jacobi theta functions

All zeros of the Jacobi theta functions are simple zeros and are given by the following:
where, are arbitrary integers.

Relation to the Riemann zeta function

The relation
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 by. The corresponding integral for is given in the article on the Hurwitz zeta function.

Relation to the Weierstrass elliptic function

The theta function was used by Jacobi to construct 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 and the constant is defined so that the Laurent expansion of at has zero constant term.

Relation to the ''q''-gamma function

The fourth theta function – and thus the others too – is intimately connected to the Jackson -gamma function via the relation

Relations to Dedekind eta function

Let be the Dedekind eta function, and the argument of the theta function as the nome. Then,
and,
See also the Weber modular functions.

Elliptic modulus

The elliptic modulus is
and the complementary elliptic modulus is

A solution to the heat equation

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions. Taking to be real and with real and positive, we can write
which solves the heat equation
This theta-function solution is 1-periodic in, and as 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 with the theta function.

Relation to the Heisenberg group

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.

Generalizations

If is a quadratic form in variables, then the theta function associated with is
with the sum extending over the lattice of integers This theta function is a modular form of weight of the modular group. In the Fourier expansion,
the numbers are called the representation numbers of the form.

Theta series of a Dirichlet character

For a primitive Dirichlet character modulo and then
is a weight modular form of level and character, which means
whenever

Ramanujan theta function

Riemann theta function

Let
the set of symmetric square matrices whose imaginary part is positive definite. is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The -dimensional analogue of the modular group is the symplectic group for, The -dimensional analogue of the congruence subgroups is played by
Then, given the Riemann theta function is defined as
Here, is an -dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with and where is the upper half-plane.
The Riemann theta converges absolutely and uniformly on compact subsets of
The functional equation is
which holds for all vectors and for all and

Poincaré series

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