Notes on Three Formulas of Abel

TL;DR

This paper examines three formulas by Abel related to Abelian integrals, analyzing their connections to residues, partial fractions, and geometric properties like genus and lattice points.

Contribution

It provides a symbolic computation perspective on Abel's formulas and clarifies their geometric and algebraic relationships.

Findings

The first two formulas are linked to residues and partial fractions.

The third formula relates to genus and lattice points in Newton polygons.

Connections between algebraic and geometric aspects of Abelian integrals are elucidated.

Abstract

These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to residues and partial fractions. The third formula arises from the first two and is related to the genus and lattice points in the Newton polygon.