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 comes from a line bundle condition of descent.
- 1 Jacobi theta function
- 2 Auxiliary functions
- 3 Jacobi identities
- 4 Theta functions in terms of the nome
- 5 Product representations
- 6 Integral representations
- 7 Explicit values
- 8 Some series identities
- 9 Zeros of the Jacobi theta functions
- 10 Relation to the Riemann zeta function
- 11 Relation to the Weierstrass elliptic function
- 12 Relation to the q-gamma function
- 13 Relations to Dedekind eta function
- 14 A solution to heat equation
- 15 Relation to the Heisenberg group
- 16 Generalizations
- 17 Notes
- 18 References
- 19 Further reading
- 20 External links
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 confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
where q = exp(πiτ) and η = exp(2πiz). It is a Jacobi form. If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity
The function also behaves very regularly with respect to its quasi-period τ and satisfies the functional equation
where a and b are integers.
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 (sometimes called theta constants.) 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 is congruent to n squared modulo 2). For the second, let
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 q, where and . 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.
The Jacobi triple product tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 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 and 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
The next two series identities were proved by István Mező
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 not zero 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 at z = 0 has zero constant term.
Relation to the q-gamma function
Relations to Dedekind eta function
See also the Weber modular functions.
A solution to heat equation
The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write
which solves the heat equation
That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:
where δ is the Dirac delta function. Thus, general solutions can be specified by convolving the (periodic) boundary condition 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.
Ramanujan theta function
Riemann theta function
be 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 analog of the congruence subgroups is played by .
Then, given , the Riemann theta function is defined as
Here, 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 where is the upper half-plane.
The Riemann theta converges absolutely and uniformly on compact subsets of
The functional equation is
which holds for all vectors , and for all and .
- Jinhee, Yi (2004), "Theta-function identities and the explicit formulas for theta-function and their applications", Journal of Mathematical Analysis and Applications 292: 381–400, doi:10.1016/j.jmaa.2003.12.009.
- 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
- Mező, István (2012). "A q-Raabe formula and an integral of the fourth Jacobi theta function". Journal of Number Theory 130 (2): 360–369.
- Abramowitz, Milton & Stegun, Irene A. (1964), Handbook of Mathematical Functions, New York: Dover Publications, ISBN 0-486-61272-4. (See section 16.27ff.)
- Akhiezer, Naum Illyich (1990) , Elements of the Theory of Elliptic Functions, AMS Translations of Mathematical Monographs 79, Providence, RI: AMS, ISBN 0-8218-4532-2.
- Farkas, Hershel M. & Kra, Irwin (1980), Riemann Surfaces, New York: Springer-Verlag, ISBN 0-387-90465-4. (See Chapter 6 for treatment of the Riemann theta)
- Hardy, G. H. & Wright, E. M. (1959), An Introduction to the Theory of Numbers (Fourth ed.), Oxford: Clarendon Press.
- Mumford, David (1983), Tata Lectures on Theta I, Boston: Birkhauser, ISBN 3-7643-3109-7.
- Pierpont, James (1959), Functions of a Complex Variable, New York: Dover.
- Rauch, Harry E. & Farkas, Hershel M. (1974), Theta Functions with Applications to Riemann Surfaces, Baltimore: Williams & Wilkins, ISBN 0-683-07196-3.
- Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- Whittaker, E. T. & Watson, G. N. (1927), A Course in Modern Analysis (Fourth ed.), Cambridge: Cambridge University Press. (See chapter XXI for the history of Jacobi's θ functions)
- Farkas, Hershel M. (2008). "Theta functions in complex analysis and number theory". In Alladi, Krishnaswami. 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 3-540-06382-X.
- Matlab code for theta function evaluation by elliptic project
This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.