
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 defined by ...
s of
several complex variables. 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 t ...
s, and
soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s. Theta functions are parametrized by points in a
tube domain inside a complex
Lagrangian Grassmannian,
namely the
Siegel upper half space.
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).
[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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
and is the
half-period ratio, confined to the
upper half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
, 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
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
for a 1-periodic
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
of . Accordingly, the theta function is 1-periodic in :
:
By
completing the square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By s ...
, 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
In mathematics, a modular form is a holomorphic function on the Upper half-plane#Complex plane, complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the Group action (mathematics), group action of the ...
, 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
''An Introduction to the Theory of Numbers'' is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. It is on the list of 173 books essential for undergraduate math libraries.
The book grew out of a series of le ...
''.
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:
:
:
:
:
Some interesting relations
If
and
, then the following theta functions
:
:
have interesting arithmetical and modular properties. When
are positive integers, then
:
:
Also if
,
, the functions with :
:
and
:
are modular forms with weight
in
i.e. If
are integers such that
,
and
there exists
,
, such that for all complex numbers
with
, we have
:
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 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 triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...
s 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:
: