NUMERICAL COMPUTATIONS OF INTEGRALS OVER PATHS ON RIEMANN SURFACES OF GENUS-N

Abstract

This paper is a continuation of work by Forest and Lee 1,2 . In 1,2 it was proved that the function theory of periodic soliton solutions occurs on the Riemann surfaces R of genus N, where the integrals over paths on R play the most fundamental role. In this paper a numerical method is developed to evaluate these integrals. Precisely, the aim is to develop a computational code for integrals of the form integral(gamma) f(z)dz/R(z), or integral(gamma) f(z)R(z)dz, where f(z) is any single-valued analytic function on the complex plane C, and R(z) is a two-valued function on C of the form GRAPHICS where {z(0)(k), 1 less than or equal to k less than or equal to 2N + delta} are distinct complex numbers which play the role of the branch points of the Riemann surface R = {(z, R(z))} of genus N - 1 + delta. The integral path gamma is continuous on R. The numerical code is developed in ''Mathematica'' 3 .