Contour integrals as $Ad$-invariant functions on the fundamental group

Abstract

We introduce a general approach to contour integrals. It covers usual Abelian integrals, the higher order Melnikov integrals and the generalized Abelian integrals. We prove that the generating function always satisfies a linear differential equation of finite order. We also present a relationship between the generalized Abelian integral and certain representation of the fundamental group of complex curve.