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34 1. Mathematical preliminaries

Thus, for example, we must regard all the terms

as when selecting terms to use for matching. We conclude this discussion with
one more example (and a few others are given in Q1.21).

E1.13 Another logarithmic example
Given

then as we write

keeping the first two terms in the asymptotic expansion, but treating ‘ln
For we write to give

if we include terms as far as To match, we write (1.73) as

retaining terms O(1), as required for (1.74). Similarly, we write (1.74) as

again retaining terms O(1), as required for (1.73), but with ‘1n it is
immediately apparent that the two expansions match (to this order): (1.75) and (1.76)
are identical (with

The processes of expanding, examining breakdowns, scaling and matching are the
essential elements of singular perturbation theory; these will provide the basis and the
framework for the rest of this text. However, there is one final aspect to which we