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when we retain the terms O(1), used in (1.69). It is clear that (1.71) and
(1.72) do not satisfy the matching principle: the term appears in one expansion
but not in the other.
However, it is easily confirmed that the two 1-term expansions [f ~ 1/(1 + x),
F ~ 1] do match, as do the 1-term/2-term expansions
Further, it is easily checked that any additional terms retained in the
expansion of by expanding the contribution, will also lead
to a failure of the matching principle. Perhaps if we retained all these terms, we might
succeed; let us therefore rewrite (1.69) as

which immediately matches with (1.70) when we treat in (1.70),
in the same way.

In order to investigate the difficulty, let us consider the three simple functions
and ln x (and we may extend this to any function which is constructed
from these elements, or can be represented as a series of such). Now introduce a new
scaled variable, X say, and so obtain, respectively,

the first two are the corresponding functions but of a different size (taking X = O(1)
and i.e. and O(1), respectively. We note, however, that the logarithmic
function does not follow this pattern: producing two terms of
different size i.e. and O(1), respectively. Indeed, we could write

but in order to match any dependence in x (or X) we would need to retain the term
ln X with (The retention of but ignoring ln x, is at the heart of the problem
we have with (1.71) and (1.72); but retaining and ln x allowed the matching
principle to be applied successfully.)
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. Indeed, the matching principle will produce an identity (as be-
fore), with the correct identification of all the individual terms involving logarithms i.e.
the interpretation is used only for the retention of the appropriate terms.