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should briefly return: the production of approximations that may prove useful in a
mainly numerical context.

1.10 COMPOSITE EXPANSIONS
When we have obtained two, or more, matched expansions that represent a function,
we are led to an intriguing question: is it possible to use these to produce, for example,
an approximate graphical representation of the function, for all x in the domain? The
main difficulty, of course, is that presumably we need to switch from one asymptotic
expansion to the next at a particular value of x–which runs counter to the matching
principle. One possibility might be to plot the function so that, for some values of
x, both matched expansions are used and allowed to overlap, but this still involves
using first one and then the other. A single expression which represents the original
function, asymptotically, for would be far preferable. Such an expression can
usually be found; it is called a composite expansion.
Suppose that we have a function which is described by two different asymptotic
expansions, and say, valid in adjacent regions, with
(We are using the notation that was introduced in example E1.11; the exten-
sion to three or more expansions follows directly.) We now introduce a function
which possesses the properties that

both as If such a function exists, it is called a composite expansion for the
original function (for obvious reasons); note that the only requirement is that the
correct behaviour, or is generated—any other (smaller) terms may not be
correct if these expansions were continued further. The issue now is how we find
we present two commonly used constructions that lead to a suitable choice for The
simpler uses a straightforward additive rule, and the other a multiplicative rule.

Definition (composite expansion—additive)

We write

where denotes the ‘overlap’ terms which are those terms involved in the match-
ing.

The inclusion of becomes obvious when we note that the terms that match must
appear in both and and so are counted twice; then removes one of them.
Given the pair and it is an elementary exercise, in particular cases, to check
that the expansion of (1.77), either for or recovers the appropriate
leading-order terms. Further, it is then possible to compare the approximation,