Theta functions

The presence of an infinitesimal, purely imaginary addition in the GF denominator turns out to be very important and to have a deep physical meaning. Its origin is related to the theta-function in Eq. (A 1.13) that represents the causality principle The reaction of a system can be caused only by perturbations at preceding instants. 

Theta functions are special functions related to Jacobian elliptic functions (Morse and Feshbach, 1953 Widder, 1975) with special properties that make then extremely useful to calculate solutions to diffusion problems for small values of time. Three of the four theta functions will be used in the present context... 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

Mumford, D., "Tata Lectures on Theta 11. Jacobian Theta Functions and Differential Equations". Birkhauser, Boston (1984). 

The theta function. This function begins with the component balance for a conventional column ... 

In the theta method, the total distillate rate and reflux ratio are all that can be specified and the theta function [Eq. (4.50)] is written to find a value of theta such that the sum of the corrected component distillate rates equals the total distillate rate ... 

Calculate the 0 (or 0 s for a complex column) and p, s such that the theta function (or functions) is very nearly zero. 

In contrast, 4<1(r) of Eq. (8.10.20) is evaluated in reciprocal space, using the theta-function transformation. Indeed, part of 4<1(r) is a periodic function, which is expanded in a Fourier series ... 

Evaluate the Poisson equation for the potential of Eq. (5.37) to determine the charge density implied. Hints relationships developed in Arfken (1985) are helpful. Additionally, the theta-function transformation (Ziman, 1972)... 

Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the... 

The feature of the Jacobian, however, which really gives it its punch is the theta function. There are 3 very good reasons to look next at the function theory of Jac -... 

In particular, there is exactly one first order theta-function, up to scalars. This important function, written i9(x), is called Riemann s theta function31. If, instead, we take any n > 3, and let ipi,..., ipng be a basis of 5n, we get ... 

Secondly, we saw in Lecture III that starting with any principal polarized abelian variety (C9 /L,H), we get Riemann s theta function d O —> C, hence 0 = (zeroes of i ) CO /L. A more succinct way to describe how P/L and H canonically determine the codimension 1 subvariety 0 up to translation34 is the following ... 

Schottky was able to show that the above identities on and implied one iden-tity on d alone, of degree 8, and Igusa has asserted that this identity holds only on (9Jt4) Moreover, when g > 4, no efficient method of eliminating from the above identities is known and the ultimate problem of characterizing t(9Rg) by simple identities in the theta-nulls remains open. I am confident that Schottky s approach has not been exhausted, however, and a full theta-function theoretic analysis of the dihedral (or even higher non-abelian) coverings of C remains to be carried out. 

The heat equation satisfied by the theta function of an abelian variety, which plays a central role in the paper of Andreotti and Mayer, has been studied by Welters in [Wl]. This paper was instrumental in Hitchin s laying the foundation of a parallel theory in the case of higher rank vector bundles on curves [H]. 

Here are references for the theory of theta functions and more generally, function theory on abelian varieties ... 

W. Baily, Classical theory of theta functions (In AMS Proc. of Symp. in pure Math., vol. 9). 

J. Fay, Theta functions on Riemann Surfaces (Springer Lecture Notes 352). 

H. Rauch, H. Farkas, Theta functions with applications to Riemann Surfaces, (Wiliams and Wilkins, 1974). 

The paper [B3] is in the Proceedings of the Summer Research Institute on Theta Functions held at Bowdoin College in 1987 which is altogether another source of references. 

Du] B.A. Dubrovin, Theta functions and non-linear equations, Russian Math. Surveys 36, 2, (1981), 11-92. 

G.E. Welters, The surface C — C on Jacobi varieties and 2nd order theta functions, Acta math. 109 (1987), 165-182. 

Meshcheryakov obtained some results on exact integration of geodesic flows of metrics PabD simple Lie groups by means of special functions. The functional nature of the solutions of the equations for geodesics is as follows they are either quasi-polynomials or rational functions of the restrictions of the theta functions of compact Riemann surfaces to rectilinear windings of Jacobian tori of these surfaces. These methods rest upon the papers by Novikov and Dubrovin [45]. 

The operators act on the space of smooth vector functions, and the Lie algebra Afn i is realized by square zero-trace matrices. The matrices a and b are diagonal with distrinct diagonal elements. According to the finite-zoned integration theory (see [77]), the commutativity equations [Lai Aa] = 0 are integrated by means of the theta-functions of the Riemann surface of the algebraic curve Q W A) = det(lV — X - Aa) = 0. 

The matrix elements of the solutions of the equations for geodesics of left-invariant metrics (pabD on the group SL(m, C) with initial data of general position turn out to be rational functions of the exponents and theta-functions of the Riemann surfaces T Q(W X) = 0. More exactly (Meshcheryakov), the solution g t) has the form g t) = e(t)r(t) exp(tZ)(/i)). The matrices e(t),r(t) are expressed here by the following formulae through the data on the Riemann surface T ... 

Corollary 4.2.3. The solutions of the Euler equations with the metrics ahD on the Lie algebra sl(m, C) are rational functions of the exponents and restrictions of the theta-functions of the algebraic curves Q(Wf A) = 0 to rectilinear windings of the Jacobian tori J T) of these curves. 

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