Extensions of asymptotic fields via meromorphic functions

An asymptotic field is a special type of Hardy field in which, module an oracle for constants, one can determine asymptotic behaviour of elements. In a previous paper, it was shown in particular that limits of real Liouvillian functions can thereby be computed. Let F denote an asymptotic field and let f is an element of F. We prove here that if G is meromorphic at the limit of f (which may be infinite) and satisfies an algebraic differential equation over R(x), then F(Gof) is an asymptotic field. Hence it is possible (module an oracle for constants) to compute asymptotic forms for elements of F(Gof). An example is given to show that the result may fail if G has an essential singularity at limf.