  By employing conformal mappings, it is possible to express the solution of
certain boundary value problems for the Laplace equation in terms of a single
integral involving the given boundary data. We show that such explicit formulae
can be used to obtain novel identities for special functions. A convenient tool
for deriving this type of identities is the so-called \emph{global relation},
which has appeared recently in a wide range of boundary value problems. As a
concrete application, we analyze the Neumann boundary value problem for the
Laplace equation in the exterior of the so-called Hankel contour, which is the
contour that appears in the definition of both the gamma and the Riemann zeta
functions. By utilizing the explicit solution of this problem, we derive a
plethora of novel identities involving the hypergeometric function.
