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Figure 2. Diagrammatic representation of the overlap region, between and

if we choose then and so, to be consistent, we should certainly
include the issue of precisely which terms we should retain will be addressed
later. Correspondingly, in the expansion (1.58), we write and
expand:

We have chosen to write down only the first three terms of this expansion, in order to
be consistent with (1.59). We see that (1.59) and (1.60) are identical and, furthermore,
this holds for all satisfying (1.56): we have verified, in this example, the rôle of the
intermediate variable. (Again, it is left as an exercise to show that the same results are
obtained directly by expanding the original function for

To proceed with this discussion, we now make choices for choice for the
O(1) expansion (e.g. (1.59)), and another for the expansion (e.g. (1.60)).
For example, in the former we might select and in the latter we use
both ‘expansions of expansions’ are valid for these choices, because both satisfy
(1.56). Thus the asymptotic expansion, constructed for x = O(1), is valid also for
correspondingly, the expansion constructed for is valid for
Hence the expansion of expansions (e.g. (1.59) or (1.60)) is valid for xs
between and the resulting expansion is now valid in an overlap region
(which is represented in figure 2).