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Jacobi's original theta function θ1 with u = iπz and with nome q = eiπτ = 0.1e0.1iπ. Conventions are (Mathematica):

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.

Contents

Jacobi theta functionEdit

 
Jacobi theta 1
 
Jacobi theta 2
 
Jacobi theta 3
 
Jacobi theta 4

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. 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.

 
Theta function θ1 with different nome q = eiπτ. The black dot in the right-hand picture indicates how q changes with τ.
 
Theta function θ1 with different nome q = eiπτ. The black dot in the right-hand picture indicates how q changes with τ.

Auxiliary functionsEdit

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

 

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

 

which is the Fermat curve of degree four.

Jacobi identitiesEdit

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

 

Then

 

Theta functions in terms of the nomeEdit

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.

Product representationsEdit

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 and w = eπiz then

 

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

 

In terms of w and q:

 

where (  ;  ) is the q-Pochhammer symbol and θ(  ;  ) is the q-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 z is real. Similar product formulas for the auxiliary theta functions are

 

Integral representationsEdit

The Jacobi theta functions have the following integral representations:

 

Explicit valuesEdit

See Yi (2004).[2] 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 identitiesEdit

The next two series identities were proved by István Mező:[3]

 

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 functionsEdit

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 functionEdit

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 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 functionEdit

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 functionEdit

The fourth theta function – and thus the others too – is intimately connected to the Jackson q-gamma function via the relation[4]

 

Relations to Dedekind eta functionEdit

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

 

and,

 

See also the Weber modular functions.

Elliptic modulusEdit

The elliptic modulus is

 

and the complementary elliptic modulus is

 

A solution to the heat equationEdit

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.[5] Taking z = x to be real and τ = it with t real and positive, we can write

 

which solves the heat equation

 

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.

Relation to the Heisenberg groupEdit

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.

GeneralizationsEdit

If F is a quadratic form in n variables, then the theta function associated with F is

 

with the sum extending over the lattice of integers   This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,

 

the numbers RF(k) are called the representation numbers of the form.

Theta series of a Dirichlet characterEdit

For   a primitive Dirichlet character modulo   and   then

 

is a weight   modular form of level   and character  , which means

 

whenever

 [6]

Ramanujan theta functionEdit

Riemann theta functionEdit

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 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

 

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  

Poincaré seriesEdit

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

NotesEdit

  1. ^ Tyurin, Andrey N. (30 October 2002). "Quantization, Classical and Quantum Field Theory and Theta-Functions". arXiv:math/0210466v1.
  2. ^ Yi, Jinhee (2004). "Theta-function identities and the explicit formulas for theta-function and their applications". Journal of Mathematical Analysis and Applications. 292 (2): 381–400. doi:10.1016/j.jmaa.2003.12.009.
  3. ^ Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
  4. ^ Mező, István (2012). "A q-Raabe formula and an integral of the fourth Jacobi theta function". Journal of Number Theory. 133 (2): 692–704. doi:10.1016/j.jnt.2012.08.025.
  5. ^ Ohyama, Yousuke (1995). "Differential relations of theta functions". Osaka Journal of Mathematics. 32 (2): 431–450. ISSN 0030-6126.
  6. ^ Shimura, On modular forms of half integral weight

ReferencesEdit

Further readingEdit

External linksEdit