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The only relevant root is X = 1 (because the other two are the ghosts of the roots
that appear for x = O(1)). To improve this approximation, we set

to give

i.e. so that a third root is as Thus the three
(real) roots are

Our introductory example, and the one above, have been rather conventional poly-
nomial equations, but the technique is particularly powerful when we have to solve,
for example, transcendental equations (which contain a small parameter). We will now
see how the approach works in a problem of this type.
E2.2 A transcendental equation
We require the approximate (real) roots, as of the equation

For x = O(1), we now have two possibilities:

but only the first option admits any roots for real, finite x. Thus x = ±1 (approxi-
mately) and then only the choice x = +1 is acceptable (because we require x > 0); a
better approximation follows directly:

The ‘expansion’ is written

where the term must be exponentially small for x = O(1) or larger (because
no roots exist if this term dominates). Now, for x > 0, there is no breakdown as