symptotic expansions are useful for
obtaining high performance analytical formulas, for validation of solutions,
acceleration of Monte-Carlo convergence, obtaining preconditioners and
adaptive grids. The references for this section are
[Erdelyi]
and
[Sveshnikov]
.
The presentation here is partial.

Definition

(O symbols) Let
be a Banach space.

1. For functions
we
write
iff
is bounded around
and
as
.

2. The series
is "asymptotic expansion" of
at
iff
We will use
notation

3. The
expansion
are called "asymptotic power series".

Proposition

(Differentiation of
asymptotic power series) If
is differentiable and both
and
possess asymptotic power series then such power series are connected by
termwise differentiation.

Proof

The proof is based on assertion that there is similar result for integration,
see the proposition (
Dominated
convergence theorem
) or less generic results from calculus. We
have
and
Thus, we must
have
Then
and we match terms with the expansion for
.