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Figure 3. Comparison of the composite expansion, equation (1.79), with the exact function
with

The agreement between the composite expansion, (1.79), and the original function,
as depicted in figure 3, is quite remarkable (and we have not chosen a particularly small
value of which would make the agreement even more pronounced). This suggests
that we should be able to obtain a formal estimate for the error when we use the
composite expansion; we will examine this conjecture in a simpler version of E1.14
(so that the manipulation is particularly routine and transparent).

E1.15 Error using a composite expansion
From E 1.14, we are given

and so we form the composite expansion