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obtained by expanding with X = O(1); correspondingly, (1.62) gives

and we see that (1.63) and (1.64) are identical, when expressed in terms of the same
variable (x or X). Further, these new expansions are not recoverable from (1.59) or
(1.60) simply by writing there: being precise about the terms to be retained
has resulted in the appearance of a new term The two expansions, (1.61) and
(1.62), 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 pertur-
bation 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. For some classes of functions, it is possible to
develop a proof which goes something like this. Define the operator which
generates the first n terms of the asymptotic expansion of as for
(written correspondingly, the operator generates the first
m terms of the asymptotic expansion of as for Here,
the two functions are identical in that for some scaling
obtained from the breakdown of the asymptotic expansion(s). Under suitable
conditions—but we are able to apply the principle more widely—it can be proved
that

when written in the same variable i.e. or (Much more on these ideas can be
found in some of the texts and references that are listed in the section on Further
Reading at the end of this chapter.) Put simply, this states that the m-term expansion of
the n-term expansion is identical to the n-term expansion of the m-term expansion;
when presented in this form, this procedure is usually associated with the name of
Milton van Dyke (1964, 1975). We present a slight variant of the principle, which
we hope is transparent and readily applicable.