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In mathematics, theta functions are special functions of several complex variables. They are important in several 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 Grassman algebra, they also appear in quantum field theory, specifically string theory and D-branes.
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 (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems). In the abstract theory this is shown to come from a line bundle condition of descent.
Jacobi theta function
The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half plane, which means it has positive imaginary part. It is given by the formula
The function also behaves very regularly with respect to addition by τ and satisfies the functional equation
where a and b are integers.
It is convenient to define three auxiliary theta functions, which we may write
This notation follows Riemann and David Mumford; Jacobi's original formulation was in terms of the nome q = exp(πτ) rather than τ, and theta there is called θ3, with termed θ0, named θ2, and called - θ1.
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. Let
The Jacobi theta function can be expressed as a product, through the Jacobi triple product theorem:
The auxiliary functions have the expressions:
The Jacobi theta functions have the following integral representations:
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 not zero 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 at z = 0 has zero constant term.
Some relations to modular forms
Let η be the Dedekind eta function. Then
As solution to heat equation
The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z=x to be real, and taking τ=it with t real and positive. Then we can write
which solves the heat equation
That this solution is unique can be seen by noting that at t=0, the theta function becomes the Dirac comb:
where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodic) boundary condition at t=0 with the theta function.
Relation to the Heisenberg group
The Jacobi theta function can be thought of as belonging to a representation of the Heisenberg group in quantum mechanics, sometimes called the theta representation. This can be seen by explicitly constructing the group. Let f(z) be a holomorphic function, let a and b be real numbers, and fix a value of τ. Then define the operators Sa and Tb such that
- (Saf)(z) = f(z + a)
- (Tbf)(z) = exp(iπb2τ + 2πibz)f(z + bτ)
but S and T do not commute:
Thus we see that S and T together with a unitary phase form a nilpotent Lie group, the (continuous real) Heisenberg group, parametrizable as where U(1) is the unitary group. A general group element then acts on a holomorphic function f(z) as
Define the subgroup as
The above theta representation of the Heisenberg group can be related to the canonical Weyl representation of the Heisenberg group as follows. Fix a value for τ and define a norm on entire functions of the complex plane as
Let be the set of entire functions f with finite norm. Note that is a Hilbert space, and that U(λ,a,b) is unitary on , and that is irreducible under this action. Then and L2(R) are isomorphic as H-modules, where H acts on L2(R) as
for and .
See also the Stone-von Neumann theorem for additional development of these ideas; note that in relation to that article, and T = expi2πQ.
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.
Riemann theta function
be set of symmetric square matrices whose imaginary part is positive definite. Hn 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 Sp(2n,Z); note that for n=1, Sp(2,Z)=SL(2,Z). The n-dimensional analog of the congruence subgroups is played by .
Then, given , the Riemann theta function is defined as
Here, is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n=1 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 .
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See section 16.27ff.)
- Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
- Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 6 for treatment of the Riemann theta)
- David Mumford, Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
- James Pierpont Functions of a Complex Variable, Dover
- Harry E. Rauch and Hershel M. Farkas, Theta Functions with Applications to Riemann Surfaces, (1974) Williams & Wilkins Co. Baltimore ISBN 683-07196-3.
This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the GFDL.
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