Computing Riemann theta functions

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision. 1. Motivation An Abelian function is a 2g-fold periodic, meromorphic function of g complex variables. Thus, Abelian functions are generalizations of elliptic functions to more than one variable. Just as elliptic functions are associated with elliptic surfaces, i.e., Riemann surfaces of genus 1, Abelian functions are associated to Riemann surfaces of higher genus. As in the elliptic case, any Abelian function can be expressed as a ratio of homogeneous polynomials of an auxiliary function, the Riemann theta function [11]. Many differential equations of mathematical physics have solutions that are written in terms of Abelian functions, and thus in terms of Riemann theta functions (see [3, 7] and references therein). Thus, to compute these solutions, one needs to compute either Abelian functions or theta functions. Whittaker and Watson [20, §21.1] state, “When it is desired to obtain definite numerical results in problems involving Elliptic functions, the calculations are most simply performed with the aid of certain auxiliary functions known as Theta-functions.” Whittaker and Watson are referring to Jacobi’s theta functions, but the same can be said for Abelian functions, using the Riemann theta function. This paper addresses the problem of computing values of the Riemann theta function and its derivatives. 2. Definition The Riemann theta function is defined by