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30 1. Mathematical preliminaries

Definition (matching principle)
We are given two asymptotic expansions of a function, defined as and valid
for x = O(1) and X = O(1), where (and either or
as in the form

respectively. These two expansions are valid in adjacent regions, so that the break-
down of one leads to the variable used in the other i.e. there are no other regions,
and associated asymptotic expansions, between them. Here, N and M are not used
simply to count the first so-many terms in the expansions; they may be used to des-
ignate the type of terms e.g. first three using the asymptotic sequence and the
first exponentially small term. However, N and M must retain these interpretations
throughout the matching process. Now we form

and

the matching principle then states that

or expressed in terms of x, if preferred. We say that the expansions ‘match to this
order’, because we can match only the terms that we have in the expansions.

Let us apply this matching principle, as we have described it, in the following example.

E1.11 An example of matching
We will show that these two expansions match:

both defined as (This is based on example E1.8.) Note that, although (1.66)
uses the first two terms in the sequence n = 0, 1, 2, ...., the expansion (1.65)