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
The presentation here is partial.
(O symbols) Let
be a Banach space.
1. For functions
is bounded around
2. The series
is "asymptotic expansion" of
We will use
are called "asymptotic power series".
asymptotic power series) If
is differentiable and both
possess asymptotic power series then such power series are connected by
The proof is based on assertion that there is similar result for integration,
see the proposition (
) or less generic results from calculus. We
Thus, we must
and we match terms with the expansion for