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 path 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 ∫γf(z)dzR(z),or∫γf(z)R(z)dz, where f(z) is any single-valued analytic function on the complex plane C, and R(z) is two-valued function on C of the form R2(z)=2N+δ∏k=1(z−z0(k)),δ=0or1, where {z0(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 genus N−1+δ. The integral path γ is continuous on ℜ. The numerical code is developed in “Mathematica” [3].
Citation:
J.-E. Lee, “Numerical computations of integrals over paths on Riemann surfaces of genus N”, TMF, 101:2 (1994), 179–188; Theoret. and Math. Phys., 101:2 (1994), 1281–1288