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48 2. Introductory applications

In this section, we will describe a technique (for equations which contain a small
parameter, as in those above) which is a natural extension of simply obtaining an
asymptotic expansion of a function, examining its breakdown, rescaling, and so on.
We will begin by examining the simple quadratic equation

and seek the solutions for The essential idea is to obtain different asymptotic
approximations for valid for different sizes of x, and see if these admit (ap-
proximate) roots. Given that we could have roots anywhere on the
real line, and so all sizes of x must be examined. (We will consider, first, only the real
roots of equations; the extension to complex roots will be discussed in due course.)
One further comment is required at this stage: we describe here a technique for find-
ing roots that builds on the ideas of singular perturbation theory. In practice, other
approaches are likely to be used in conjunction with ours to solve particular equations
e.g. sketching or plotting the function, or using a standard numerical procedure (such
as Newton-Raphson). There is no suggestion that this expansion technique should be
used in isolation—it is simply one of a number of tools available.
Returning to (2.1), if x = O(1), then

and so we have a root x = – 1 (approximately). In order to generate a better ap-
proximation, we may use any appropriate method. For example, we could invoke the
familiar procedure of iteration, so we may write

with Then we obtain

and so on (but note that iteration may not generate a correct asymptotic expansion at
a given order in It is clear from this approach that a complete representation of the
root will be obtained if we use the asymptotic sequence and so an alternative is
to seek this form directly—and this is more in keeping with the ideas of perturbation
theory. Thus we might seek a root in the form

so that (2.1) can be written as