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56 2. Introductory applications

First, we expand for x = O(1) and for to give

and

both for we have retained terms as far as in each expansion. (You
should confirm that these two expansions satisfy the matching principle.)
Now these two expansions are valid in the overlap region, represented by
defined by

thus we express the integral as

The only requirement, at this stage, is that we are able to perform the integration of the
various functions that appear in the asymptotic expansions. Note that the first integral
has been expressed as an integration in X—the most natural choice of integration
variable in this context. To proceed, we obtain

and this is to be expanded for and (note!). Thus we obtain