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In
mathematics, theta functions are
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defin ...
s of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
. They show up in many topics, including
Abelian varieties,
moduli spaces,
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
solitons. 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 a ...
.
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 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'' usually ...
).
[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
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasio ...
) 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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...
and
Mumford;
Jacobi's original formulation was in terms of the
nome rather than . In Jacobi's notation the -functions are written:
:
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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, 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:
:
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
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...
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
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 expansion c ...
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
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
, 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 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
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
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
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
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \c ...
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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
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 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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
. 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 order ...
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
*
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Elliptic functions
Riemann surfaces
Analytic functions
Several complex variables