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(where the use of ‘= 0’ here is to imply ‘equal to zero to all orders in thus
and so on. We have one root

It is clear that (2.2) admits only one root; the other—it is a quadratic equation that we
are solving—must appear for a different size of x.
The ‘asymptotic expansion’ (we treat the function as such)

remains valid for and so there is no new root for x = o(1); however, this
expansion does break down where We define
and write

or

This approximation admits the roots X = 0 and X = –1, so now the quadratic equa-
tion has a total of three roots! Of course, this cannot be the case; indeed, it is clear that
the root X = 0 is inadmissible, because the ‘asymptotic expansion’
breaks down where (which is x = O(1) and so returns us to (2.5)). The only
available root is X = –1, and this is the second (approximate) root of the equation
(leaving X = 0 as no more than a ‘ghost’ of the root x ~ –1). The expansion for F
does not further breakdown (as and so there are no other roots—not that we
expected any more! We may seek a better approximation, as we did before, in the form

which gives

i.e. thus

or

The two roots of the quadratic equation, (2.1), are therefore