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6 1. Mathematical preliminaries

we describe in this text. If we now consider small, for X fixed, the size of x is now
proportional to and the results are very different:

indeed, in this example, we cannot even write down a suitable approximation of (1.14)
for small The expression in (1.14) attains a maximum at X= 1/2, and for larger X
the function tends to zero.
We observe that any techniques that we develop must be able to handle this situation;
indeed, this example introduces the important idea that the function of interest may
take different (approximate) forms for different sizes of x. This, ultimately, is not
surprising, but the significant ingredient here is that ‘different sizes’ are measured in
terms of the small parameter, We shall be more precise about this concept later.

E1.3 Another simple second-order equation
This time we consider

with
(The use of here, rather than is simply an algebraic convenience, as will become
clear; obviously any small positive number could be represented by or —or anything
equivalent, such as or et cetera.) Presumably—or so we will assume—a first
approximation to equation (1.15), for small is just

but this problem has no solution. The general solution is where A and
B are the two arbitrary constants, and no choice of them can satisfy both conditions.
In a sense, this is a more worrying situation than that presented by either of the two
previous examples: we cannot even get started this time.
The exact solution is

and the difficulties are immediately apparent: with x fixed, gives
but then how do we accommodate the condition at infinity? Correspondingly, with
and fixed, we obtain and now how can we obtain the dependence
on As we can readily see, to treat and x separately is not appropriate here—we
need to work with a scaled version of x (i.e. The choice of such a variable
avoids the non-uniform limiting process: and