Liouville theorem with parameters: asymptotics of certain rational integrals in differential fields

TL;DR

This paper investigates the asymptotic behavior of parameter-dependent rational integrals in differential fields, extending Liouville's classical theorem through formal power series and determinant identities, with applications.

Contribution

It introduces a parametric version of Liouville's theorem and develops methods using formal power series and Vandermonde determinants for analyzing rational integrals.

Findings

Derived asymptotic formulas for parameter-dependent integrals

Established identities from Vandermonde determinant expansions

Extended classical Liouville theorem to parametric setting

Abstract

We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients following from generalized Vandermonde determinant expansion. Our result can be viewed as a parametric version of a classical theorem of Liouville. We also give applications.