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can be satisfied if either, but not necessarily both, and are O(1). Let us see how
this arises in an example.

E2.3 An equation with complex roots
Here, using the more usual notation for a complex number, we consider

and immediately we obtain

and so we have roots approximately. Thus we write

and so the equation becomes

This is satisfied if etc., and thus we have two complex roots

and we observe that the imaginary part is O(1), but that the real part is
The full ‘expansion’ is clearly not uniformly valid as there is a breakdown
where or We introduce and write

and so

which produces the single, available root near Z = – 1 and then, more accurately, we
have The equation has three roots, two of which are complex:

Finally, a class of equations for which this direct approach (for complex roots) is not use-
ful is characterised by the appearance of terms such as (or anything equivalent).