Numerical computations of integrals over paths on Riemann surfaces of genusN

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 genusN, where the integrals over paths on ℜ play the most fundamental role. In this paper a numerical method is developed to evaluate these integrals. Predisely, the aim is to develop a computational code for integrals of the form

$$\int\limits_\gamma {f(z)\frac{{dz}}{{R(z)}}, or} \int\limits_\gamma {f(z)R(z)dz,} $$

wheref(z) is any single-valued analytic function on the complex planeC, andR(z) is a two-valued function onC of the form

$$R^2 (z) = \prod\limits_{k = 1}^{2N + \delta } {(z - z_0 (k)), \delta = 0 or 1,} $$

where {z ₀(k),1≤k≤2N+δ} are distinct complex numbers which play the role of the branch points of the Riemann surface ℜ = {(z, R(z))} of genusN−1+δ. The integral path γ is continuous on ℜ. The numerical code is developed in “Mathematica” [3].