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and so we have the set of (approximate) roots

A few examples of other equations with complex roots (some of which may be real,
of course) are set as exercises in Q2.5.

2.2 INTEGRATION OF FUNCTIONS REPRESENTED
BY ASYMPTOTIC EXPANSIONS
Our second direct, and rather routine application of these ideas is to the evaluation
of integrals. In particular, we consider integrals of functions that are represented by
asymptotic expansions in a small parameter; this may involve one or more expansions,
but if it is the latter—and it often is—then the expansions will satisfy the matching
principle.
The procedure that we adopt calls upon two general properties: the first is the
existence of an intermediate variable (valid in the overlap region; see §1.7), and the
second is the familiar device of splitting the range of integration, as appropriate. We
then express the integral as a sum of integrals over each of the asymptotic expansions
of the integrand, the switch from one to the next being at a point which is in the
overlap region. The expansions are then valid for each integration range selected and,
furthermore, the value of the original integral (assuming that it exists) is independent
of how we split the integral. Thus the particular choice of intermediate variable is
unimportant; indeed, it may be quite general, satisfying only the necessary conditions
for such a variable; see (1.56), for example. Let us apply this technique to a simple
example.

E2.5 An elementary integral
We are given

and we require the value, as of the integral

(Note that the integral here is elementary, to the extent that it may be evaluated directly,
although we will integrate only the relevant asymptotic expansions; this example has
been selected so that the interested reader may check the results against the expansion
of the exact value.)