In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, theta functions are
special functions of
several complex variables. They show up in many topics, including
Abelian varieties,
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s,
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s, and
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
s. As
Grassmann algebras, they appear in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
.
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 ), 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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
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
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' twig'' usuall ...
).
[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
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to b ...
.
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
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
, which means it has 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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
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 could 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 Jacobi may refer to:
* People with the surname Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenvalue algorithm, ...
'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. 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
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
, 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 In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several iden ...
) 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:
:
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
:
Note that the following modular identities hold:
:
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:
:
Some series identities
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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, 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
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by ...
also, since
:
where the second derivative is with respect to and the constant is defined so that the
Laurent expansion
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
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 relation
:
Relations 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 is
:
and the complementary elliptic modulus is
:
A solution to the heat equation
The Jacobi theta function is the
fundamental solution of the one-dimensional
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
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 In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heis ...
of the Heisenberg group.
Generalizations
If is a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
in variables, then the theta function associated with is
:
with the sum extending over the
lattice of integers
. This theta function is a
modular form of weight (on an appropriately defined subgroup) of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
. 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 and character
:
which means
[Shimura, On modular forms of half integral weight]
:
whenever
:
Ramanujan theta function
Riemann theta function
Let
:
the set of
symmetric square
matrices whose imaginary part is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
.
is called the
Siegel upper half-space and is the multi-dimensional analog of the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
. The -dimensional analogue of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
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
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
. 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.
Theta function coefficients
If and are positive integers, any arithmetical function and , then
:
The general case, where and are any arithmetical functions, and is
strictly increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
with , is
:
Notes
References
*
*
*. ''(for treatment of the Riemann theta)''
*
*
*
*
*
* ''(history of Jacobi's functions)''
Further reading
*
*
*
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, .
External links
*
{{Authority control
Elliptic functions
Riemann surfaces
Analytic functions
Several complex variables