Theta function on arvau

Wikipedia / 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: "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.

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

so that the sum converges.

This analytic function can be used to solve a combinatorics

This is a generating function is not a perfect square, there are zero ways.

Application to elliptic functions

Theta functions occur most commonly in the theory of elliptic functions, 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".

(in order to resolve issues of choice of branch).

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 .

, the function is an entire function on the complex plane, so by Liouville's theorem 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:

with the nome . 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 p-adic numbers.

Product representations

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers and with and

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 q-Pochhammer symbol and 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 is real. Similar product formulas for the auxiliary theta functions are

, 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:

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:

shall be displayed:

Explicit values

:Wiktionary:lemniscaticLemniscatic 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,

and Dedekind eta function

If the reciprocal of the Gelfond constant values can be represented in a simplified way by using the hyperbolic lemniscatic sine:

the Lemniscate constant is represented.

Note that the following modular identities hold:

is the Rogers–Ramanujan continued fraction:

EquianharmonicEquianharmonic 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(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:

the now mentioned formula is valid.

And for this formula two examples shall be given:

inserted: inserted: represents the golden ratio 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:

is Golden ratio number again.

Infinite sum of the reciprocals of the Pell numbers

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

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 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 q-gamma function via the relation

Relations to Dedekind eta function

Let be the Dedekind eta function, and the argument of the theta function as the nome . Then,

and,

Elliptic modulus

The elliptic modulus

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

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 write

which 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 is

with the sum extending over the lattice. This theta function is a modular form of weight (on an appropriately defined subgroup) 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 then

is a weight modular form of level which means

whenever

Ramanujan theta function

Riemann theta function

Let

be 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

, the Riemann theta function is defined as

is an -dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with 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 functional equation is

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:

this formula of the Euler beta function is valid:

Exemplary elliptic integrals

In the following some Elliptic Integral Singular Values are derived:

Combination of the integral identities with the nome

For the proof of the correctness of these nome values, see the article Nome (mathematics)!

On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:

Partition sequences and Pochhammer products

Regular partition number sequence

itself indicates the number of ways in which a positive integer number with all associated number partitions are listed in the following table:

The summandization of the now mentioned Pochhammer product is described by the pentagonal number theorem in this way:

The following basic definitions apply to the pentagonal numbers and the card house numbers:

As a further application one obtains a formula for the third power of the Euler product

Strict partition number sequence

can be split into positive integer summands such that each summand appears at most once and no summand value occurs repeatedly. Exactly the same sequence is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:

Overpartition number sequence

The Maclaurin series for the reciprocal of the function has the numbers of over partition sequence as coefficients with a positive sign:

, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number .

First example:

These 14 possibilities of partition markings exist for the sum 4:Second example:

These 24 possibilities of partition markings exist for the sum 5:

Relations of the partition number sequences to each other

are even.

can be written with the regular partition sequence P and the strict partition sequence Q In the following table of sequences of numbers, this formula should be used as an example:Related to this property, the following combination of two series of sums can also be set up via the function :

Notes

References

. (for treatment of the Riemann theta)
(history of Jacobi's functions)