By R. Sneider

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62), are said to match to this order (because we can match only the terms available in the original expansions). The matching principle is a fundamental tool in the techniques of singular perturbation theory; it is invoked, sometimes as a check, but more often as a means for determining arbitrary constants (or functions) that are generated in the solution of differential equations. Although we have not presented the matching principle as a proven property of functions—it is one reason why we call it a ‘principle’—we have every confidence in its validity.

55)—which is sufficient. 55) may seem superfluous, and it is in a strictly numerical sense, but it contains important information about the nature of the underlying function (and it helps us better to understand the breakdown). Because we are interested in the behaviour of functions (as and not simply numerical estimates, we shall retain such terms when they provide useful and relevant information. 26 1. 7 INTERMEDIATE VARIABLES AND THE OVERLAP REGION In our examples thus far, we have expanded the given functions for x = O(1), and, in one case, for We now investigate other scalings which correspond to sizes that sit between those generated by the breakdown of an asymptotic expansion.

We must emphasise, however, that the thrust 16 1. Mathematical preliminaries of this text is towards the introduction of methods which aid the description of the structure of a solution (in the limit under consideration). Finally, before we move on, we briefly comment on functions of a complex variable. ) Given and the limit we are able to construct asymptotic expansions exactly as described above, but with one important new ingredient. e. take the limit, from any direction whatsoever. g. for (for some and for other args the asymptotic expansion (with the same asymptotic sequence, fails because for some n.