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36 1. Mathematical preliminaries

against the original function, by estimating for we will
return to this shortly. An example of the additive rule is now presented.

E1.14 A composite expansion (additive)
From E1.8 (see equations (1.52), (1.53) and (1.55)), we have

and

where and the matching involves (see E 1.11)

Following (1.77), we define

and then we obtain directly

retaining O(1) and terms. Correspondingly, for X = O(1), we have

retaining terms O(l) and Thus, to this order, (1.79) is a composite expansion
for the function (1.78), for To confirm the accuracy of this approximation,
we compare (1.78) and (1.79), for in figure 3.