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.
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 (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(πiτ) is the nome and η = exp(2πiz). 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
The function also behaves very regularly with respect to its quasi-period τ and satisfies the functional equation
where a and b are integers.
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 (or half-period) functions are defined by
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
which is the Fermat curve of degree four.
Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/ to z (n ≡ n2 mod 2). For the second, let
Theta functions in terms of the nome
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 and w = eπiz then
We therefore obtain a product formula for the theta function in the form
In terms of w and q:
which we may also write as
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:
See Yi (2004). Edit: Proper credit for these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so the results below are either in Ramanujan's lost notebook or follow immediately from it.
Some series identities
These relations hold for all 0 < q < 1. Specializing the values of q, 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 m, n are arbitrary integers.
Relation to the Riemann zeta function
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.
Relation to the Weierstrass elliptic 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.
Relation to the q-gamma function
The fourth theta function – and thus the others too – is intimately connected to the Jackson q-gamma function via the relation
Relations to Dedekind eta function
See also the Weber modular functions.
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 z = x to be real and τ = it with t real and positive, we can write
which solves the heat equation
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.
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.
If F is a quadratic form in n variables, then the theta function associated with F is
the numbers RF(k) 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
Ramanujan theta function
Riemann theta function
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 n-dimensional analogue of the modular group is the symplectic group for n = 1, The n-dimensional analogue of the congruence subgroups is played by
Then, given the Riemann theta function is defined as
The Riemann theta converges absolutely and uniformly on compact subsets of
The functional equation is
which holds for all vectors and for all and
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