By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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**Example text**

E. there are no other regions, and associated asymptotic expansions, between them. g. first three using the asymptotic sequence and the first exponentially small term. However, N and M must retain these interpretations throughout the matching process. Now we form and the matching principle then states that or expressed in terms of x, if preferred. We say that the expansions ‘match to this order’, because we can match only the terms that we have in the expansions. Let us apply this matching principle, as we have described it, in the following example.

The device that we therefore adopt, when log terms are present, is to treat for the purposes of retaining the relevant terms; the matching principle, as we stated it, is then valid. e. the interpretation is used only for the retention of the appropriate terms. 34 1. Mathematical preliminaries Thus, for example, we must regard all the terms as when selecting terms to use for matching. 21). 74). 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.

13); of course, we cannot perform the integration, but we can 18 1. Mathematical preliminaries generate a suitable approximation via the familiar technique of integration by parts. In particular we obtain and so on, to give Note that we have used a standard mathematical procedure, which has automatically generated a sequence of terms—indeed, it has generated an asymptotic sequence, defined as This is another important observation: our definitions have implied a selection of the asymptotic sequence, but in practice a particular choice either appears naturally (as here) or is thrust upon us by virtue of the structure of the problem; we will write more of this latter point in due course.