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52 2. Introductory applications

thus any other roots that might exist must arise as Indeed, as
we see that the expansion (2.8) breaks down where and so we set
to give

This approximation has one root at X = 0, but this cannot be used directly because
the expansion, (2.9), itself breaks down as This occurs where
i.e. so a further scaling must be introduced: to produce

Since we have

we have a root near to obtain an improved approximation, we write

and so obtain

As there is no further breakdown, and so we have found two real roots

A number of other equations, both polynomial and transcendental, are discussed in the
exercises Q2.2, 2.3 and 2.4. However, all these involve the search for real roots; we now
turn, therefore, to a brief discussion of the corresponding problem of finding all roots,
whether real or complex. It will soon become clear that we may often adopt precisely
the same approach when any roots are being sought (although, sometimes, there may
be an advantage in writing and working with two, coupled, real equations).
The only small word of warning is that the size of the real and imaginary parts, measured
in terms of may be different e.g. x = O(1) now implies that which