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

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 coefficients of this series following from the generalized Vandermonde determinant expansion. Our result can be viewed as a parametric version of a classical theorem of Liouville. We also give some applications.