Integration of algebroidal functions

TL;DR

This paper introduces the concept of integrating algebroidal functions on Riemann surfaces, establishing properties, conditions for path independence, and showing that integrals of irreducible functions remain irreducible under certain conditions.

Contribution

It is the first to define and analyze the integration of algebroidal functions on Riemann surfaces, including residue-based path independence criteria.

Findings

Integration of algebroidal functions is well-defined on Riemann surfaces.

Path independence is characterized by residues at critical points.

Integrals of irreducible algebroidal functions are also irreducible if residues vanish.

Abstract

In this paper, we introduce the integration of algebroidal functions on Riemann surfaces for the first time. Some properties of integration are obtained. By giving the definition of residues and integral function element, we obtain the condition that the integral is independent of path. At last, we prove that the integral of an irreducible algebroidal function is also an irreducible algebroidal function if all the residues at critical points are zeros.