In mathematics, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.
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 z), 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.
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 z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
It also turns out to be τ-quasiperiodic in z, with
Thus, in general,
for any integers a and b.
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
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 z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. Alternatively, we obtain four functions of q only, defined on the unit disk . They are sometimes called theta constants:[note 1]
with q = eiπτ.
These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
which is the Fermat curve of degree four.
Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n ≡ n2 mod 2). For the second, let
Theta functions in terms of the nome
Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q, where w = eπiz and q = eπiτ. In this form, the functions become
We see that the theta functions can also be defined in terms of w and q, 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.
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 q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then
We therefore obtain a product formula for the theta function in the form
In terms of w and q:
which we may also write as
This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
The Jacobi theta functions have the following integral representations:
Some series identities
These relations hold for all 0 < q < 1. Specializing the values of q, 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 m, n are arbitrary integers.
Relation to the Riemann zeta function
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 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 z and the constant c is defined so that the Laurent expansion of ℘(z) at z = 0 has zero constant term.
Relation to the q-gamma function
Relations to Dedekind eta function
See also the Weber modular functions.
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 z = x to be real and τ = it with t real and positive, we can write
which solves the heat equation
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 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.
If F is a quadratic form in n variables, then the theta function associated with F is
the numbers RF(k) are called the representation numbers of the form.
Theta series of a Dirichlet character
For χ a primitive Dirichlet character modulo q and ν = 1 − χ(−1)/2 then
is a weight 1/2 + ν modular form of level 4q2 and character
Ramanujan theta function
Riemann theta function
the set of symmetric square matrices whose imaginary part is positive definite. ℍn is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The n-dimensional analogue of the modular group is the symplectic group Sp(2n,ℤ); for n = 1, Sp(2,ℤ) = SL(2,ℤ). The n-dimensional analogue of the congruence subgroups is played by
Then, given τ ∈ ℍn, the Riemann theta function is defined as
Here, z ∈ ℂn is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n = 1 and τ ∈ ℍ where ℍ 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 τ 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 ℂn × ℍn.
The functional equation is
which holds for all vectors a, b ∈ ℤn, and for all z ∈ ℂn and τ ∈ ℍn.
Generalized theta functions
There are in general higher-order non-quadratic theta functions. They have the form
where q = e2πiz. The variable z lies in the upper half plane, χ(n) is any arithmetical function and κ is an integer greater than 1.
The restriction κ = 2 and
with χ a Dirichlet character is related to the classicaly theta series of a Dirichlet character θχ(z). Some properties are
Here ε(n) = μ(n)/n when n ≠ 0, and zero otherwise. The arithmetical function μ(n) is the Moebius μ function. The arithmetical functions nκ(n) and n∗
κ(n) are evaluated from the prime factorization of n by decomposing it into a power of κ and its κ-free part. This is as follows:
Here, bk1, bk2,..., bks and cj1, cj2,..., cjs are positive integers with all cj < κ. This decomposition is unique and by convection we set
The function n∗
κ(n) can also evaluated as
when nκ(n) ≠ 0. Setting
then Cκ(χ;n) is multiplicative whenever χ(n) is multiplicative.
Another property is
Theta function coefficients
If a and b are positive integers, χ(n) any arithmetical function and |q| < 1, then
- for all with .
- Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. sec. 16.27ff. ISBN 978-0-486-61272-0.
- Akhiezer, Naum Illyich (1990) . Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs. 79. Providence, RI: AMS. ISBN 978-0-8218-4532-5.
- Farkas, Hershel M.; Kra, Irwin (1980). Riemann Surfaces. New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8.. (for treatment of the Riemann theta)
- Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford: Clarendon Press.
- Mumford, David (1983). Tata Lectures on Theta I. Boston: Birkhauser. ISBN 978-3-7643-3109-2.
- Pierpont, James (1959). Functions of a Complex Variable. New York: Dover Publications.
- Rauch, Harry E.; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams & Wilkins. ISBN 978-0-683-07196-2.
- Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
- Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's θ functions)
- Farkas, Hershel M. (2008). "Theta functions in complex analysis and number theory". In Alladi, Krishnaswami (ed.). Surveys in Number Theory. Developments in Mathematics. 17. Springer-Verlag. pp. 57–87. ISBN 978-0-387-78509-7. Zbl 1206.11055.
- Schoeneberg, Bruno (1974). "IX. Theta series". Elliptic modular functions. Die Grundlehren der mathematischen Wissenschaften. 203. Springer-Verlag. pp. 203–226. ISBN 978-3-540-06382-7.
- Ackerman, M. Math. Ann. (1979) 244: 75. "On the Generating Functions of Certain Eisenstein Series" Springer-Verlag
- Moiseev Igor. "Elliptic functions for Matlab and Octave".
- Tyurin, Andrey N. (30 October 2002). "Quantization, Classical and Quantum Field Theory and Theta-Functions". arXiv:math/0210466v1.
- Yi, Jinhee (2004). "Theta-function identities and the explicit formulas for theta-function and their applications". Journal of Mathematical Analysis and Applications. 292 (2): 381–400. doi:10.1016/j.jmaa.2003.12.009.
- Proper credit for 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 the results below are either in Ramanujan's lost notebook or follow immediately from it.
- Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
- Mező, István (2012). "A q-Raabe formula and an integral of the fourth Jacobi theta function". Journal of Number Theory. 133 (2): 692–704. doi:10.1016/j.jnt.2012.08.025.
- Ohyama, Yousuke (1995). "Differential relations of theta functions". Osaka Journal of Mathematics. 32 (2): 431–450. ISSN 0030-6126.
- Shimura, On modular forms of half integral weight
- Nikolaos D. Bagis, "q-Series Related with Higher Forms". arXiv:2006.16005v4 [math.GM] 10 Mar 2021, https://arxiv.org/pdf/2006.16005.pdf