I'm looking for a mathematically rigorous overview of asymptotic expansions. These pop up all over the place and physicists have several heuristic tricks for dealing with them, but these tricks tend to be imprecisely stated and unproven. For example, in John Boyd's overview of the subject he states the "Optimal Truncation Rule" as
For a given $\epsilon$, the minimum error in an asymptotic series is usually achieved by truncating the series so as to retain the smallest term in the series, discarding all terms of higher degree.
(Boldface added by me.)
Is there a good resource that develops a theory of asymptotic expansions with rigorous definitions and theorems?
Everything I have found so far has been either a physicist's handbook of imprecise heuristics, or a catalog of results without proofs for the already-familiar.
Classics are A. Erdelyi's "Asymptotic Expansions", E.T. Copsons's "Asymptotic Expansions", N.G. de Bruijn's "Asymptotic methods in analysis", Bleistein-et-al's "Asymptotic expansions of integrals", and Dingle's "Asymptotic expansions..."
The first three of these are available as inexpensive paperbacks.
Basic asymptotics of integrals, with proofs and examples, are also treated in my note http://www.math.umn.edu/~garrett/m/v/basic_asymptotics.pdf
Asymptotics of ODE's at regular and certain irregular singular points are linked to at various places on the notes page http://www.math.umn.edu/~garrett/m/mfms/