Theta function
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
- Physics
- Geometry
Theta functions in two dimensions are functions of two complex arguments. In one choice of parameter, for example, z encodes position on a two-dimensional lattice, and τ or q 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 z and q are complex numbers and |q| < 1 so that the sum converges.
This analytic function can be used to solve a combinatorics problem: in how many different ways can each integer n be written as the sum of two squares?
When z=0, this function becomes
This is a generating function where the coefficient on represents how many ways there are to write k as a perfect square—when k=0, there is just one way. When k is any other perfect square, there are two ways:. When k is not a perfect square, there are zero ways.
If you square this generating function, you obtain. If you collect terms by exponent, you find that is a generating function where the coefficient on counts how many ways there are to write k as the sum of any two squares. This count includes negative integers and order, such that,, and each count as separate ways of making 32 + 42 = 25.
Application to elliptic functions
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 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 .
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 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:
[Image:Complex theta animated1.gif|500px|thumb|center|Theta function with different nome. The black dot in the right-hand picture indicates how changes with.]
[Image:Complex theta animated2.gif|500px|thumb|center|Theta function with different nome. The black dot in the right-hand picture indicates how changes with.]
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 (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:
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, letThen
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 becomeWe 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 haveIt 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
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">:Wiktionary:lemniscatic">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. 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 nome 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 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 sum 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 relationwas 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, sincewhere 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 relationRelations 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 isand the complementary elliptic modulus is
Derivatives of theta functions
These are two identical definitions of the complete elliptic integral of the second kind:The derivatives of the Theta Nullwert functions have these MacLaurin series:
The derivatives of theta zero-value functions are as follows:
The two last mentioned formulas are valid for all real numbers of the real definition interval:
And these two last named theta derivative functions are related to each other in this way:
The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:
For the derivation of these derivation formulas see the articles Nome (mathematics) and Modular lambda function!
Integrals of theta functions
For the theta functions these integrals are valid:The final results now shown are based on the general Cauchy sum formulas.
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 writewhich 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 positive-definite quadratic form in variables, then the theta function associated with iswith 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 thenis a weight modular form of level and character
which means
whenever
Riemann theta function
Letbe 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. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking to be the period matrix with respect to a canonical basis for its first homology group.
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.Derivation of the theta values
Identity of the Euler beta function
In the following, three important theta function values are to be derived as examples:This is how the Euler beta function is defined in its reduced form:
In general, for all natural numbers this formula of the Euler beta function is valid:
Exemplary elliptic integrals
In the following some Elliptic Integral Singular Values are derived:The ensuing function has the following lemniscatically elliptic antiderivative: For the value this identity appears: This result follows from that equation chain:
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