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26 1. Mathematical preliminaries

1.7 INTERMEDIATE VARIABLES AND THE OVERLAP REGION
In our examples thus far, we have expanded the given functions for x = O(1),
and, in one case, for We now investigate other scalings which correspond
to sizes that sit between those generated by the breakdown of an asymptotic expansion.
This will lead us to an important and significant principle in the theory of singular
perturbations.
Let us suppose that we have an asymptotic expansion of a function which is valid for
x = O(1), and another of the same function which is valid for further, the
breakdown of at least one of these expansions produces the scaling used in the other.
The line we now pursue is to examine what happens to these expansions when we
allow where

i.e. the size (scale) of x is smaller than O(1) but not as small as Given that the
expansion valid for x = O(1) breaks down at the asymptotic ordering of the
terms is unaltered if we use i.e. it is still valid for this size of x. Conversely,
we are given that the expansion valid for breaks down where x = O(1), but
it remains valid for x smaller than O(1)—so this is also valid for Hence both
expansions are valid for intermediate variable; furthermore, this validity
holds for all which satisfy (1.56). In order to make plain what is happening here,
let us apply this procedure to an example.

E1.9 Example with an intermediate variable
We are given the two asymptotic expansions

both as (It is left as an exercise to show that these expansions are obtained
from the function but we do not need to know
the form of the function in what follows.)
In the expansion (1.57), we write where is defined in (1.56), and
expand:

where we have retained terms O(1), and It is not clear how many terms
we should retain, without being more precise about the size of For example,