A generalized jacobi theta function and quasimodular forms masanobu kaneko and don zagier in this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by robbert dijkgraaf 1, theorem 2 and corollary. In fact, we can choose any form of the function fat t 0 fq,q,0 fq,q 22 and then 21 will. If the polar angle is extended to the complex plane, the jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. Chazys equation is solved by modular forms of level 1. First we analyze convergence of the theta functions. Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with jacobi theta function, in english.
It is also one of the main tools in the study of the spectrum of the laplace operator, and is thus of some auxiliary importance throughout mathematical physics. Properties of jacobi thetafunction the jacobi theta. It is getting a bit far from the material of this course, but i wanted to. Using the heat kernel you sum over a lattice of periods to match the periodicity of the theta function. The four principal theta functions are defined by the following rapidly. The jacobi theta function is the fundamental solution of the onedimensional heat equation with. This gives rise to the action of the modular group on theta functions. Thus, there is a theta function attached to any abelian variety. On the growth of the jacobi theta function stack exchange. We can start only from halphens equation and the heat equation in order to study theta functions form analytic vieqpoint. One jacobi theta function named after carl gustav jacob jacobi is a.
The jacobi theta functions,, and satisfy the following parity type relations with respect to. Etale theta part 02 properties of the arithmetic jacobi. Differential equations involving cubic theta functions and eisenstein. Theta functions are entire functions whose quotients are elliptic functions. It is shown, for example, that riemanns formula, a fundamental degree 4 identity, can be. Mathematical function, suitable for both symbolic and numerical manipulation. Jacobis construction is based on an application of thetafunctions cf. Setting this issue aside temporarily, we move to a problem of optimal control to show another area in which the equation arises naturally. The application of fourier series to the heat conduction on a circular ring is considered. Chapter 9 the functional equation for the riemann zeta. Jacobis thirdorder ode satisfied by the theta null functions.
Inside the unit q disk, and have branch cuts from to. Elliptic function resulting from the direct inversion of elliptic integrals cf. Etale theta part 02 properties of the arithmetic jacobi theta function. Derivation of the functional equation of the jacobi theta function. Jacobi see 4, 5, although his notation is distinct from the following notation. Addition formulas for jacobi theta functions, dedekinds eta function, and ramanujans congruences article pdf available in pacific journal of mathematics 2401 march 2009 with 423 reads. The theta function satisfies the following heat equation. The heisenberg group and theta functions with 5 characteristics 4. I recommend the book a brief introduction to theta functions by richard bellman reprinted by dover publications.
Mathworld and wikipedia dont give a good english reference, at least from what i skimmed over. There are various methods to derive this functional equation, see e. Definitions illustrations series and product representations zeros specific values argument transformations lattice transformations sums and products differential equations integrals representation of other functions representation by other functions approximations. Given any integer m0, let r m denote the number of elements x2 such that xx 2m. Abelian differential, thus determining the jacobi variety of the riemann surface and an associated theta function. One considers more general theta functions, involving characteristics. Other property such as addition formulae can be deduced from 0.
Hamilton jacobi theory december 7, 2012 1 free particle thesimplestexampleisthecaseofafreeparticle,forwhichthehamiltonianis h p2 2m andthehamiltonjacobiequationis. Solving those is rather tedious, and we use loose upper bounds to simplify. Applying poisson summation for lattices gives the desired result. The jacobi triple product identity gives a closed form for many in nite product generating functions that arise naturally in combinatorics and number theory. In mathematics, theta functions are special functions of several complex variables. These basic prop erties of and follo w from a represen tation of 2 as the mellin transform of a function in v olving deriv ativ es of jacobis theta function. In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. In the nal section, we present some modern theory of the equation. Abelian differential, thus determining the jacobi variety of the riemann. By the poisson summation formula the numbertheoretic theta function. Some generalizations of the rigidity theorems and some general. Zeta function part 10 jacobi theta function youtube. Corollary each homeomorphism of the riemann surface induces a unitary map on theta functions.
Jacobi elliptic functions encyclopedia of mathematics. We can start only from halphens equation and the heat equation in order to. Periods of modular forms and jacobi theta functions 451 g2 drops out and thus whose taylor coefficients in each degree are modular forms in r. Apr 01, 2016 in this video we talk about proposition 1. Pdf addition formulas for jacobi theta functions, dedekind.
The jacobi theta functions and are the periodic functions with respect to with period and a quasi. We construct elliptic genera of level 1 in section 4, for both spin manifolds and almost complex manifolds with c 1. This came out of conversations with emmanuel lepage. Contemporary mathematics heat kernels, theta identities, and zeta. A generalized jacobi theta function, mathematical methods in. Several special values of jacobi theta functions istva. Notes on the poisson summation formula, theta functions, and. Titchmarsh, the theory of the riemann zeta function. David mumfords tata lectures on theta volumes are an excellent source of reading on this topic. The other jacobi theta functions,, and are even functions with respect to.
Applications of the jacobi theta functions include the analytic solution of the heat equation, square potential well problems in quantum mechanics, wannier functions in solid state physics, conformal mapping of periodic regions, gravitational physics, quantum cosmology, coding theory, sphere packings, crystal lattice calculations, and study of the fractional quantum hall effect. A generalized jacobi theta function, mathematical methods. This is the one whose mellin transform is the riemann zeta function, see at riemann zeta function relation to jacobi theta function. Doubly periodic meromorphic functions via kz,t 24 7. Thus, there is a thetafunction attached to any abelian variety. At the end of the paper, we show how is it possible to extend our arguments and deduce a wide. Periods of modular forms and jacobi theta functions. Rhoades1 department of mathematics, stanford university, stanford, ca 94305 edited by george e.
Read a generalized jacobi theta function, mathematical methods in the applied sciences on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Hamiltonjacobi theory december 7, 2012 1 free particle thesimplestexampleisthecaseofafreeparticle,forwhichthehamiltonianis h p2 2m. This section is about a more general theta function, called the jacobi theta function. Quintic polynomial solution by jacobi theta function. Theta functions in general are powerfully versatile. It is getting a bit far from the material of this course, but i wanted to write it up here so that you can see the connection to the heat and schr odinger equations on the circle. The unit circle is the natural boundary of the region of analyticity for all jacobi theta functions,,,, and. Chapter 9 the functional equation for the riemann zeta function.
But this writeup will briefly describe how the theta function solves the heat equation. But now m is onehalf and the addition of which we have before the function equation of our quasiperiodic function plays no role in the proof. The bestknown properties and formulas for the jacobi theta functions elliptictheta. But this writeup will brie y describe how the theta function solves the heat equation. Then we prove that any suitably wellbehaved solution of this equation must coincide with the in. Notes on the poisson summation formula, theta functions. The jacobi theta function is the fundamental solution of the onedimensional heat equation with spatially periodic boundary conditions. Generic hjb equation the value function of the generic optimal control problem satis es the hamiltonjacobibellman equation. This is a try to solve a problem of landweber in la1. Jacobi theta function article about jacobi theta function.
Jacobi theta inversion formula and deduce two formulas for the associ ated zeta. A nonlinear differential equation related to the jacobi. Fourier series, kernel for heat conduction along a circular ring and jacobi theta function jambunatha sethuraman physics department vinayaka mission s kirupananda variyar engineering college, salem, tamil nadu, india 636308 abstract. The gluing construction of lattices is used to generate and study a number of theta function densities. We give a proof based on a functional equation for the jacobi theta function z p 1. The solution of this differential equation can be expressed in terms of the jacobi elliptic function dn u,k. Evaluatingthetaderivativeswithrational characteristics arxiv. The first expansion you wrote is the fourier series of the theta function. In mathematics, particularly qanalog theory, the ramanujan theta function generalizes the form of the jacobi theta functions, while capturing their general properties. In particular, the jacobi triple product takes on a particularly elegant form when written in terms of the ramanujan theta. In particular, the conditions are satisfied by the canonical period matrix for abelian differentials of the first kind on a riemann surface cf. This is clear since were counting lattice points on the surface of a sphere of radius 2min ndimensions. Gauss intergral complex integration with maximum length inequaltiy. The jacobi theta functions,, and are odd functions with respect to.
So according to this theorem, according to the proof of this theorem in this case, the jacobi theta series has only one zero in any fundamental domain of our lattice. In section 3, we will study theta functions using halphens equation. A nonlinear differential equation for the polar angle of a point of an ellipse is derived. Jacobi s construction is based on an application of theta functions cf. In this section we will show this, in the next use the functional equation of the theta function to give a functional equation for the zeta function. Evaluatingthetaderivativeswithrational characteristics.