Numerical computations of integrals over paths on Riemann surfaces of genus $N$

J.-E. Lee

Teoretičeskaâ i matematičeskaâ fizika, Tome 101 (1994) no. 2, pp. 179-188

Résumé

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 $\Re$ of genus $N$, where the integrals over path on $\Re$ 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 $$ \displaystyle \int _{\gamma }\,f(z)\frac {dz}{R(z)},\qquad \text {or}\qquad \displaystyle \int _{\gamma }\, f(z)R(z)\,dz,$$ where $f(z)$ is any single-valued analytic function on the complex plane $\mathbf C$, and $R(z)$ is two-valued function on $\mathbf C$ of the form $$ R^2(z)=\displaystyle \prod ^{2N+\delta }_{k=1}\,(z-z_0(k)),\qquad \delta =0\quad \text {or}\quad 1,$$ where $\bigl \{z_0(k),1\le k\le 2N+\delta \bigr \}$ are distinct complex numbers which play the role of the branch points of the Riemann surface $\Re =\bigl \{(z,R(z))\bigr \}$ of genus $N-1+\delta$. The integral path $\gamma$ is continuous on $\Re$. The numerical code is developed in “Mathematica” [3].