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Encyclopedia > Theta function

In mathematics, theta functions are special functions of several complex variables. They are important in several 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, specifically string theory and D-branes. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, several functions are important enough to deserve their own name. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... In theoretical physics, D-branes are a special class of p-branes, named for the mathematician Johann Dirichlet. ...


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. In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ...

Contents

Jacobi theta function

The Jacobi theta function 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 In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...

vartheta_{00}(z; tau) = sum_{n=-infty}^infty exp (pi i n^2 tau + 2 pi i n z) = 1 + 2 sum_{n=1}^infty left(e^{pi itau}right)^{n^2} cos(2pi n z).

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 Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ...

vartheta_{00}(z+1; tau) = vartheta_{00}(z; tau).

The function also behaves very regularly with respect to its quasi-period τ and satisfies the functional equation

vartheta_{00}(z+a+btau;tau) = exp(-pi i b^2 tau -2 pi i b z),vartheta(z;tau)

where a and b are integers.


Auxiliary functions

It is convenient to define three auxiliary (or half-period) theta functions, which may be written

begin{align} vartheta_{01}(z;tau)& = vartheta!left(z+{textstylefrac{1}{2}};tauright)[3pt] vartheta_{10}(z;tau)& = exp!left({textstylefrac{1}{4}}pi i tau + pi i zright) vartheta!left(z + {textstylefrac{1}{2}}tau;tauright)[3pt] vartheta_{11}(z;tau)& = exp!left({textstylefrac{1}{4}}pi i tau + pi i!left(z+{textstyle frac{1}{2}}right)right)vartheta!left(z+{textstylefrac{1}{2}}tau + {textstylefrac{1}{2}};tauright). end{align}

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = exp(πiτ) rather than τ. In Jacobi's notation the θ-functions are written like this: Bernhard Riemann. ... David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

begin{align} vartheta_{00}(z;tau)& = theta_3(z,q) quad& vartheta_{01}(z;tau)& = theta(z,q)[3pt] vartheta_{10}(z;tau)& = theta_2(z,q) quad& -vartheta_{11}(z;tau)& = theta_1(z,q). end{align}

The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions - notational variations for further discussion. There are a number of notational systems for the Jacobi theta functions. ...


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 A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...

vartheta_{00}(0;tau)^4 = vartheta_{01}(0;tau)^4 + vartheta_{10}(0;tau)^4

which is the Fermat curve of degree four. Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ...


Jacobi identities

Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ+1 and τ ↦ -1/τ. We already have equations for the first transformation; for the second, let In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

alpha = (-i tau)^{frac{1}{2}} exp!left(frac{pi}{tau} i z^2 right).,

Then

begin{align} vartheta!left({textstylefrac{z}{tau}; frac{-1}{tau}}right)& = alpha,vartheta(z; tau)quad& vartheta_{01}!left({textstylefrac{z}{tau}; frac{-1}{tau}}right)& = alpha,vartheta_{10}(z; tau)[3pt] vartheta_{10}!left({textstylefrac{z}{tau}; frac{-1}{tau}}right)& = alpha,vartheta_{01}(z; tau)quad& vartheta_{11}!left({textstylefrac{z}{tau}; frac{-1}{tau}}right)& = -alpha,vartheta_{11}(z; tau) end{align}

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 = exp(πiz) and q = exp(πiτ). In this form, the functions become

begin{align} vartheta_{00}(w, q)& = sum_{n=-infty}^infty (w^2)^n q^{n^2}quad& vartheta_{01}(w, q)& = sum_{n=-infty}^infty (-1)^n (w^2)^n q^{n^2}[3pt] vartheta_{10}(w, q)& = sum_{n=-infty}^infty (w^2)^{left(n+frac{1}{2}right)} q^{left(n + frac{1}{2}right)^2}quad& vartheta_{11}(w, q)& = i sum_{n=-infty}^infty (-1)^n (w^2)^{left(n+frac{1}{2}right)} q^{left(n + frac{1}{2}right)^2} end{align}

So 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...


Product representations

The Jacobi triple product tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have In mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi theta function, normally written as a series, as a product. ...

prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 + w^{2}q^{2m-1}right) left( 1 + w^{-2}q^{2m-1}right) = sum_{n=-infty}^infty w^{2n}q^{n^2}.

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 = exp(πiτ) and w = exp(πiz) then

vartheta(z; tau) = sum_{n=-infty}^infty exp(pi i tau n^2) exp(pi i z 2n) = sum_{n=-infty}^infty w^{2n}q^{n^2}.

We therefore obtain a product formula for the theta function in the form

vartheta(z; tau) = prod_{m=1}^infty left( 1 - exp(2m pi i tau)right) left( 1 + exp((2m-1) pi i tau + 2 pi i z)right) left( 1 + exp((2m-1) pi i tau -2 pi i z)right)

Expanding terms out, the Jacobi triple product can also be written

prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 + (w^{2}+w^{-2})q^{2m-1}+q^{4m-2}right),

which we may also write as

vartheta(z|q) = prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 + 2 cos(2 pi z)q^{2m-1}+q^{4m-2}right).

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

vartheta_{01}(z|q) = prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 - 2 cos(2 pi z)q^{2m-1}+q^{4m-2}right).
vartheta_{10}(z|q) = 2 q^{1/4}cos(pi z)prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 + 2 cos(2 pi z)q^{2m}+q^{4m}right).
vartheta_{11}(z|q) = -2 q^{1/4}sin(pi z)prod_{m=1}^infty left( 1 - q^{2m}right) left( 1 - 2 cos(2 pi z)q^{2m}+q^{4m}right).

Integral representations

The Jacobi theta functions have the following integral representations:

vartheta_{00} (z; tau) = -i int_{i - infty}^{i + infty} {e^{i pi tau u^2} cos (2 u z + pi u) over sin (pi u)} du
vartheta_{01} (z; tau) = -i int_{i - infty}^{i + infty} {e^{i pi tau u^2} cos (2 u z) over sin (pi u)} du.
vartheta_{10} (z; tau) = -i e^{iz + i pi tau / 4} int_{i - infty}^{i + infty} {e^{i pi tau u^2} cos (2 u z + pi u + pi tau u) over sin (pi u)} du
vartheta_{11} (z; tau) = e^{iz + i pi tau / 4} int_{i - infty}^{i + infty} {e^{i pi tau u^2} cos (2 u z + pi tau u) over sin (pi u)} du

Relation to the Riemann zeta function

The relation

vartheta(0;-1/tau)=(-itau)^{1/2} vartheta(0;tau)

was used by Riemann to prove the functional equation for Riemann's zeta function, by means of the integral Bernhard Riemann. ... In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...

Gammaleft(frac{s}{2}right) pi^{-s/2} zeta(s) = frac{1}{2}int_0^inftyleft[vartheta(0;it)-1right] t^{s/2}frac{dt}{t}

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. In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...


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 In mathematics, Weierstrasss elliptic functions are a standard type of elliptic functions (the other is the Jacobis elliptic functions). ...

wp(z;tau) = -(log vartheta_{11}(z;tau))'' + c

where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of wp(z) at z = 0 has zero constant term. In mathematics, a Laurent series is an infinite series. ...


Some relations to modular forms

Let η be the Dedekind eta function. Then The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...

vartheta(0;tau)=frac{eta^2left(frac{tau+1}{2}right)}{eta(tau+1)}.

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 The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

vartheta (x,it)=1+2sum_{n=1}^infty exp(-pi n^2 t) cos(2pi nx)

which solves the heat equation

frac{partial}{partial t} vartheta(x,it)=frac{1}{4pi} frac{partial^2}{partial x^2} vartheta(x,it).

That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb: In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ш). From the orthogonality of the Fourier series...

lim_{trightarrow 0} vartheta(x,it)=sum_{n=-infty}^infty delta(x-n)

where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodic) boundary condition at t = 0 with the theta function. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...


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. In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ... In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. ...


Generalizations

If F is a quadratic form in n variables, then the theta function associated with F is In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

theta_F (z)= sum_{min Z^n} exp(2pi izF(m))

with the sum extending over the lattice of integers Zn. This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion, See lattice for other meanings of this term, both within and without mathematics. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, the modular group &#915; (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

theta_F (z) = sum_{k=0}^infty R_F(k) exp(2pi ikz),

the numbers RF(k) are called the representation numbers of the form.


Ramanujan theta function

See main articles Ramanujan theta function and mock theta function.

In mathematics, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. ... A mock theta function is one of certain special functions written down by Srinivasa Ramanujan, in his last letter to G. H. Hardy and in his lost notebook. ...

Riemann theta function

Let

mathbb{H}_n={Fin M(n,mathbb{C}) ; mathrm{s.t.}, F=F^T ;textrm{and}; mbox{Im} F >0 }

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 textrm{Ker} {textrm{Sp}(2n,mathbb{Z})rightarrow textrm{Sp}(2n,mathbb{Z}/kmathbb{Z}) }. Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a Siegel upper half-plane is the set of n×n symmetric matrices over the complex number field whose imaginary part is positive definite. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, the modular group &#915; (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. ...


Then, given tauin mathbb{H}_n, the Riemann theta function is defined as

theta (z,tau)=sum_{min Z^n} expleft(2pi i left(frac{1}{2} m^T tau m +m^T z right)right).

Here, zin mathbb{C}^n 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 tau in mathbb{H} where mathbb{H} is the upper half-plane. In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...


The Riemann theta converges absolutely and uniformly on compact subsets of mathbb{C}^ntimes mathbb{H}_n.


The functional equation is

theta (z+a+tau b, tau) = exp 2pi i left(-b^Tz-frac{1}{2}b^Ttau bright) theta (z,tau)

which holds for all vectors a,b in mathbb{Z}^n, and for all z in mathbb{C}^n and tau in mathbb{H}_n.


Q-theta function

See main article Q-theta function.

In mathematics, the q-theta function is a type of q-series. ...

References

  • Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See section 16.27ff.)
  • Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
  • Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 6 for treatment of the Riemann theta)
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fourth edition (1959) , Oxford University Press
  • David Mumford, Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
  • James Pierpont Functions of a Complex Variable, Dover
  • Harry E. Rauch and Hershel M. Farkas, Theta Functions with Applications to Riemann Surfaces, (1974) Williams & Wilkins Co. Baltimore ISBN 0-683-07196-3.
  • E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927. (See chapter XXI for the history of Jacobi's θ functions)

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Sir Edward Maitland Wright (February 13, 1906 - February 2, 2005) was an English mathematician, and collaborator with G.H. Hardy. ... David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ... (George) Neville Watson (31 January 1886 - 2 February 1965) was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. ...

External Links

  • Abramowitz and Stegun hosted by Simon Fraser University

This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
PlanetMath: functional equation for the theta function (166 words)
The lemma used in the derivation of the functional equation for the Riemann Xi function.
This functional equation is not as remarkable as the one for the Xi function, because it does not actually extend the domain of the function.
This is version 7 of functional equation for the theta function, born on 2003-01-29, modified 2006-11-25.
  More results at FactBites »

 
 

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