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(1.42) hold for If the answer is ‘yes’, then the expansion is said to be regular
or uniform or uniformly valid; if not, then the expansion is singular or non-uniform or
not uniformly valid. Further, it is not unusual to use the terms breakdown or blow up to
describe the failure of an asymptotic expansion. To explore these ideas, we introduce
a first, simple example.

E1.6 An example of
Let us consider the function

for and use the binomial expansion to obtain the ‘natural’ asymptotic expan-
sion, valid for x = O(1):

Here, the asymptotic sequence is and we have taken the expansion as far as terms
at But the domain of f is given as and clearly the expansion (1.44) is
not even defined on x = 0 (which is more dramatic than simply not being valid near
x = 0). Thus (1.44) is not uniformly valid–indeed, it ‘blows up’ at x = 0.
The original function can, of course, be evaluated at x = 0:

and now another complication is evident. The asymptotic sequence used in (1.44)
does not include terms and so it could never give the correct value on
x = 0, even if the terms were defined there. Clearly, the expansion in (1.44) has been
obtained by treating x large relative to but this cannot be true if x is sufficiently
small. The critical size is where x is about the size of which is precisely the idea that
led us to the introduction of a scaled version of x. Let us write then

where we have labelled the same function, expressed in terms of X and as
The binomial expansion of (1.46), for with X = O(1), yields

which, on X = 0, recovers (1.45).