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54 2. Introductory applications

In this case, it is almost always convenient to formulate the problem in real and imag-
inary parts, and the appearance of a small parameter does not affect this approach in
any significant way; we present an example of this type.

E2.4 A real-imaginary problem
We seek all the roots of the equation

as note that

and so the form of this problem does indeed exhibit this more complicated structure.
Let us write z = x + iy, and then (2.9) becomes

or

We see immediately that the right-hand sides of these two equations do not presage a
breakdown of these contributions, as x or y increases or decreases; thus we proceed
with x = O(1) and y = O(1). Now equation (2.10b) possesses the solutions

and this is the relevant choice (rather than y = 0) because we require sin x cosh y > 1
(from (2.10a)). Then equation (2.10a) gives

and this is consistent only if n = 2m (m = 0, ±1, ±2, ...) because cosh y > 0. Finally,
the solution arises only for since cosh as we see that a
solution exists where and so we introduce Thus we obtain