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To conclude this discussion, we briefly describe the multiplicative rule which can
be used as an alternative for the construction of a composite expansion. So, with the
same notation as introduced earlier:

Definition (composite expansion—multiplicative)
Now we write

provided that for

Here, the terms involved in the matching appear in both and and so
dividing by cancels one of these. It is clear, therefore, that a composite expansion, to
a given order, is not unique; indeed, any number of variants exists—we have presented
two possible choices only. We will provide an example of the application of (1.82), but
we are unable to use the expansions quoted in E1.14 (which would have been useful
as a comparison) because, there, However,
a slightly simpler version of E1.14 is possible.

E1.16 A composite expansion (multiplicative)
See E1.14; let us be given

and

the matching of these expansions involves simply Thus, (1.82) gives

and so

and

thus (1.83) is a (multiplicative) composite expansion for (1.78). (An examination of
the accuracy of this expansion can be found in Q1.24, and a few additional examples
of composite expansions appear in Q1.22, 1.23.)