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24 1. Mathematical preliminaries

in separable form i.e. for any choice of the functions a and Thus we must
extend the definition of an asymptotic expansion to accommodate this:

Definition (asymptotic expansion with a parameter 2)
We write (cf. (1.42)

for x = O(1) and every where

as

It is clear that the separable case is simply a special version of this more general defi-
nition; let us investigate an example which incorporates such a term.

E1.8 One more example of
Consider the function

for with x = O(1) we obtain

because the term is exponentially small. This asymptotic expansion, (1.53),
as written down, is uniformly valid: there is no breakdown as and the asymptotic
ordering of the terms is even reinforced as However, from (1.52), we see that

which is not the result we obtain from (1.53): the (complete) expansion, started in
(1.53), cannot be uniformly valid! Of course, it is clear that the difficulty is associated
with the exponential term; it is this which contributes to the boundary value (on
x = 0), but it is ignored in the 2-term asymptotic expansion (1.53).
The rôle of the exponential term becomes evident when we retain it, following our
familiarscaling which gives