In mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the behavior of discrete multi-dimensional periodic systems, such as crystal lattices or points on a torus. Because they are smooth, they allow the study and manipulation of discrete combinatorial systems using the tools of analysis.
For this reason, theta functions have useful applications in topics such as
number theory: "in how many ways can a number be written as a sum of squares?" physics: "how does heat flow on a toroidal ring?", "how do quantum particles behave when arranged in a lattice?" geometry: "what are the shape properties of elliptic curves?"
and others, including abelian varieties, moduli spaces, quadratic forms, and solitons.
Theta functions in two dimensions are functions of two complex arguments. In one choice of parameter, for example, encodes position on a two-dimensional lattice, and or encodes the shape of the lattice. In higher dimensions, the shape of the lattice is dictated by a matrix; in general, theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space.
Basic example One example of a theta function is
where and are complex numbers and so that the sum converges.
This analytic function can be used to solve a combinatorics problem: in how many different ways can an integer be written as the sum of two squares? When , we have
This is a generating function where the coefficient of represents how many ways there are to write as a perfect square: when , there is just one way. When is any other perfect square, there are two ways: . When is not a perfect square, there are zero ways.
Squaring this generating function, we obtain
Collecting terms by exponent, we find that is a generating function where the coefficient of counts how many ways there are to write as the sum of any two squares. This count includes negative integers and order, such that , , and : each count as separate ways of making .
Application to elliptic functions Theta functions occur most commonly 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. Abstractly, this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".
Throughout this article, should be interpreted as (in order to resolve issues of choice of branch).See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation when is outside the strip . Here, denotes the principal branch of the complex logarithm.
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 and , where can be any complex number and is the half-period ratio, confined to the upper half-plane, which means it has a positive imaginary part. It is given by the formula
where is the nome and . It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed , this is a Fourier series for a 1-periodic entire function of . Accordingly, the theta function is 1-periodic in :
By completing the square, it is also -quasiperiodic in , with
Thus, in general,
for any integers and .
For any fixed , the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in unless it is constant, and so the best we can do is to make it periodic in and quasi-periodic in . Indeed, since and , the function is unbounded, as required by Liouville's theorem.
It is in fact the most general entire function with 2 quasi-periods, in the following sense:
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 (or half-period) 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 (notational variations) for further discussion.
If we set in the above theta functions, we obtain four functions of only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of only, defined on the unit disk . They are sometimes called theta constants: for all with .
with the nome . Observe that . These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
or equivalently,
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 (a special case of the Macdonald identities) 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 (noting some authors instead set ) 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
In particular, so we may interpret them as one-parameter deformations of the periodic functions , again validating the interpretation of the theta function as the most general 2 quasi-period function.
Integral representations The Jacobi theta functions have the following integral representations:
The Theta Nullwert function as this integral identity:
This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
Furthermore, the theta examples and shall be displayed:
Explicit values Lemniscatic values
Proper credit for most of 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 they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004). Define,
with the nome and Dedekind eta function Then for
If the reciprocal of the Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding values or values can be represented in a simplified way by using the hyperbolic lemniscatic sine:
With the letter the Lemniscate constant is represented.
Note that the following modular identities hold:
where is the Rogers–Ramanujan continued fraction:
Equianharmonic values The mathematician Bruce Berndt found out further values of the theta function:
Further values Many values of the theta function and especially of the shown phi function can be represented in terms of the gamma function:
Nome power theorems
Direct power theorems
For the transformation of the nomeAndreas Dieckmann: Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta. Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021. in the theta functions these formulas can be used:
The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the Pythagorean triples according to the Jacobi identity. Furthermore, those transformations are valid:
These formulas can be used to compute the theta values of the cube of the nome:
And the following formulas can be used to compute the theta values of the fifth power of the nome:
Transformation at the cube root of the nome The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
Transformation at the fifth root of the nome The Rogers-Ramanujan continued fraction can be defined in terms of the Jacobi theta function in the following way:
The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
Modulus dependent theorems
In combination with the elliptic modulus, the following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
And this is an efficient formula for the cube of the nome:
For all real values the now mentioned formula is valid.
And for this formula two examples shall be given:
First calculation example with the value inserted:
Second calculation example with the value inserted:
The constant represents the golden ratio number exactly.
Some series identities
Sums with theta function in the result
The infinite sumLandau (1899) zitiert nach Borwein, Page 94, Exercise 3. of the reciprocals of Fibonacci numbers with odd indices has the identity:
By not using the theta function expression, following identity between two sums can be formulated:
Also in this case is Golden ratio number again.
Infinite sum of the reciprocals of the Fibonacci number squares:
Infinite sum of the reciprocals of the Pell numbers with odd indices:
Sums with theta function in the summand 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