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28 1. Mathematical preliminaries

We have made one choice but the same argument can be developed
for other choices; in particular, we could use As p decreases,
so the overlap region extends; indeed, we may allow this to proceed provided p > 0
(because (1.56) still holds). Of course, if we permit the limit then conditions
(1.56) will be violated, although we may allow p to be as close as we desire to zero. This
obviously prompts the question: what does happen to our procedure—the expansion of
expansions—if we do select p = 0? After all—being naïve—it would seem but a small
step from p nearly zero (which is permitted) to p = 0 (although we are all aware that
there can be big differences between and x = a in some contexts!). In fact, this
situation here is not unfamiliar; it is analogous to the discussion that must be undertaken
when the convergence of a series is investigated. Given that a series is convergent for
and divergent for its status for (i.e. the two
cases x = a ± R) must be investigated via individual and special calculations. Here, we
will employ the same philosophy, namely, to apply our procedure in the case p = 0,
and note the results; they may, or may not, prove useful. In the event, it will transpire
that the results are fundamentally important, and lead to a very significant property of
asymptotic expansions.

1.8 THE MATCHING PRINCIPLE
Again, we suppose that we have two asymptotic expansions, one valid for x = O(1) and
one for exactly as described in the previous section. This time, however,
we expand the first expansion for and the second for x = O(1), i.e. the
overlap region is the maximum that we can envisage (and one step beyond anything
permitted so far). We know that this procedure is acceptable for the pair
with 0 < p < 1, but now we set p = 0. Let us investigate this by returning to our
previous example.

E1.10 Example with the maximum overlap
As in E1.9, we are given

and we expand (1.61) further, using We retain terms O(1) and because
we have no information about terms and smaller. Thus we obtain