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is a small complication here: is embedded in the second boundary condition, (3.5).
In order to use our familiar methods, we must first reformulate this condition.
We assume (and this must be checked at the conclusion of the calculation) that
on can be expanded as a Taylor series about r = a i.e.

Now the problem—albeit via a boundary condition—contains the parameter in
a form which suggests that we may seek a solution for based on the asymptotic
sequence Thus we write

and then

with

and from (3.6):

and so on. The problem for is precisely that for the undistorted circle, so

as given in (3.1).
The problem for now becomes that of finding a solution of Laplace’s equa-
tion which satisfies

and as The most natural way to proceed is to represent as a
Fourier Series, and then a solution for follows directly by employing the method
of separation of variables. As a particularly simple example of this, let us suppose that
and so

the relevant solution is then