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22 1. Mathematical preliminaries

Thus we have two representations of one valid for x = O(1), (1.44), and
one for (1.47). Further, the latter expansion is defined on X = 0 (i.e. x = 0)
and gives the correct value (as an expansion of With these observations in
place, we are now in a position to discuss uniformity and breakdown more completely
and more carefully.

1.6 UNIFORMITY OR BREAKDOWN
Suppose that we wish to represent for by an asymptotic expansion

which has been constructed for x = O(1). This expansion is uniformly valid if

for every and Conversely, it breaks down (and is therefore non-uniform)
if there is some and some such that

In other words, the expansion is said to break down if there is a size of x, in the
domain of the function, for which two consecutive terms in the asymptotic expansion
are the same size. On the other hand, the expansion is uniformly valid if the asymptotic
ordering of the terms, as represented by the asymptotic sequence is maintained
for all x in the domain.
It is an elementary exercise to apply this principle to our previous example; from
(1.44) we have

and the domain of the original function is As the second term in the
expansion, (1.48), becomes the same size as the first where the expansion
has broken down. That is, for x of this size, the expansion (1.48) is no longer valid;
in order to determine the form of the expansion for we must return to the
function and use this choice i.e. write is exactly how we generated
(1.47). Thus the breakdown of an expansion can lead us to the choice of a new,
scaled variable, and we note that this is based on the properties of the expansion, not
any additional or special knowledge about the underlying function. (This point is
important for what will come later: when we solve differential equations, we will not
have the exact solution available—only an asymptotic expansion of the solution. But,
as we shall see, the equation itself does hold information about possible scalings.) We
apply this principle of breakdown and rescaling to another example.