Testing for asymptotic series

Question

I would like to know if there is a systematic way of writing down an asymptotic series representation of a function? (like one can use Taylor series expansion for doing a power series).

Conversely given a series and a function is there a way to decide whether it is an asymptotic series of the function?

Most examples I see in books are kind of very special where one could develop an asymptotic series because of some special properties of the integral representation of the function like for the $Ei(x)$ etc. You basically integrate by parts and and in those special cases it works out.

Or if there is some simple test that one can do on a function to test whether it has an asymptotic expansion about that point?

Answer 1

Look into the work of Bruno Salvy (especially the joint work with John Shackell), you will find a lot of decision procedures for these types of problems. If you want even more, then you can dig into Joris van der Hoeven's work on the same topic.

The answer to your question is essentially: if your function is defined from a regular-enough process, then usually yes the problem is solvable (up to solvability of the problem in your ground field), otherwise it usually is not. The many papers of the authors above make that comment of mine extremely precise, I'm just relaying the intuition.

Most of Salvy's and van der Hoeven's work consists of both theory and effective implementations of their results in (usually) publicly available code [although Salvy's work is generally coded in Maple, so you need a license for that, but most universities do already].