Jacobi ϑ functions

The Jacobi ϑ functionsMathworldPlanetmath are 4 basic functions of Jacobi’s theory of elliptic functionsMathworldPlanetmath. They are functions of two complex variables, z and τ, known as the argumentMathworldPlanetmath and the half-period ratio, respectively. It is often convenient to use the quantity q=eπiτ, which is known as the nome. When the half-period ratio is used, these functions are denoted ϑj(z|τ) (the index j runs from 1 to 4), and when the nome is used, the functions are denoted ϑj(z;q). q and τ are sometimes snipped for brevity when they are obvious from the context, and when that is done, the functions are denoted ϑj(z).

These functions can be defined by the following series. It is also possible to express them as products and as integralsDlmfPlanetmath (http://planetmath.org/IntegralRepresetationsOfJacobiVarthetaFunctions) — see the attachments to this entry for details.

ϑ1(z;q)=n=-(-1)nq(n+1/2)2e(2n+1)iz=2n=0(-1)nq(n+1/2)2sin(2n+1)z (1)
ϑ1(zτ)=n=-(-1)neiπτ(n+1/2)2+(2n+1)iz (2)
ϑ2(z;q)=n=-q(n+1/2)2e(2n+1)iz=2n=0q(n+1/2)2cos(2n+1)z (3)
ϑ2(zτ)=n=-eiπτ(n+1/2)2+(2n+1)iz (4)
ϑ3(z;q)=n=-qn2e2inz=1+2n=1qn2cos(2nz) (5)
ϑ3(zτ)=n=-eiπτn2+2inz (6)
ϑ4(z;q)=n=-(-1)nqn2e2inz=1+2n=1(-1)nqn2cos(2nz) (7)
ϑ4(zτ)=n=-(-1)neiπτn2+2inz (8)

Note that these series converge for all complex values of z whenever |q|<1 (equivalently, when τ>0). Furthermore, these series converge uniformly on compact subsets (this may be shown using the Weierstrass M-testMathworldPlanetmathPlanetmath) so these functions are analytic.

The theta functions satisfy many identitiesPlanetmathPlanetmath, the most important of which are the quasiperiodicity identities (http://planetmath.org/QuasiperiodAndHalfquasipreiodRelationsForJacobiVarthetaFunctions), Jacobi’s identity (http://planetmath.org/JacobisIdentityForVarthetaFunctions), and Landen’s transformation. For these identities and others, please see the attachments.

The theta functions were originally introduced because it is possible to express the Jacobi elliptic functionsMathworldPlanetmath as ratios of theta functions. In some ways, this role is similarMathworldPlanetmath to the role the complex exponentialMathworldPlanetmathPlanetmath plays in the theory of trigonometric functionsDlmfMathworld. Just as one can derive complicated trigonometric identities form the properties of the exponential functionsDlmfDlmfMathworldPlanetmath, so too one can derive complicated identites for elliptic functions using the properties of theta functions.

They are very useful in the numerical analysis of elliptic functions, since the series given above converge rapidly. Hence (as was realized early on by Jacobi), it is usually better to compute elliptic functions by first computing theta functions.

In addition, theta functions are interesting in their own right and appear in numerous, often surprising contexts, as the following exampes show. Theta functions appear as Green’s functions for the heat equation. In number theoryMathworldPlanetmath, they are used to study the representations of integers as sums of squares. Theta functions can be used to construct modular functionsDlmfMathworld. They can be used to construct integral representations of generating functions. In theoretical physics, they are used to perform sums over crystals and describe hexagonal lattices of vortices.

Title Jacobi ϑ functions
Canonical name JacobivarthetaFunctions
Date of creation 2013-03-22 14:08:10
Last modified on 2013-03-22 14:08:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 21
Author rspuzio (6075)
Entry type Definition
Classification msc 33E05
Defines Jacobi theta functions
Defines nome