# New approach to certain real hyper-elliptic integrals

## TL;DR

The paper introduces a new basis for hyper-elliptic integrals, providing explicit formulas and symmetries that simplify calculations and reduce the number of formulas needed, improving computational efficiency.

## Contribution

A novel basis for hyper-elliptic integrals is developed, enabling explicit formulas and revealing symmetries that simplify their computation.

## Key findings

- Explicit formulas for hyper-elliptic integrals derived.
- Symmetries from dihedral group actions identified.
- Reduction in the number of formulas needed for computations.

## Abstract

In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis ${\phi}_n(x)$ of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formulas for the considered integrals. Our approach, specified to elliptic case, is more effective than known recursive procedures for elliptic integrals. We also show that basic integrals enjoy symmetry coming from the action of the dihedral group $D_{M}$ on a real projective line. This action is closely connected with the properties of homographic transformation of a real projective line. This explains similarities occurring in some formulas in popular tables of elliptic integrals. As a consequence one can reduce the number of necessary formulas in a significant way. We believe that our results will simplify programming and computing the hyper-elliptic integrals in various problems of mathematical physics and engineering.