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32 1. Mathematical preliminaries

In our presentation of the matching principle we have described how it can be
applied to functions which involve and exponential terms (or functions which can
be expanded in terms of these). However, when we apply this same procedure to
logarithmic functions, we encounter a difficulty which requires a careful adjustment
of the matching principle.

1.9 MATCHING WITH LOGARITHMIC TERMS
To see that we have a problem, let us consider an appropriate example, expand and
then attempt to match in the way described above.

E1. 12 A logarithmic example
We are given

and we construct the asymptotic expansions for x = O(1) and as
So we obtain

where we have written down the first two terms. (Note that as
Correspondingly, we have

where we have retained, again, the first two terms. We now match (1.69) and (1.70),
and so we write (1.69) in terms of X and expand:

retaining terms O(1), as required for (1.70). Similarly, from (1.70), we obtain