Symbolic Integration in Weierstrass-like Extensions

TL;DR

This paper explores symbolic integration within Weierstrass-like differential field extensions, introducing new algorithms and formulas for integrals involving functions defined by nonlinear differential equations.

Contribution

It extends classical special polynomials to Weierstrass-like extensions and develops algorithms for reduction and integration in these complex fields.

Findings

New algorithms for reduction in Weierstrass-like extensions

Formulas for integrals of powers of the Weierstrass $ ext{wp}$ function

Enhanced understanding of integration in nonlinear differential fields

Abstract

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\wp$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of $\wp$.