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By employing conformal mappings, it is possible to express the solution ofcertain boundary value problems for the Laplace equation in terms of a singleintegral involving the given boundary data. We show that such explicit formulaecan be used to obtain novel identities for special functions. A convenient toolfor 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 aconcrete application, we analyze the Neumann boundary value problem for theLaplace equation in the exterior of the so-called Hankel contour, which is thecontour that appears in the definition of both the gamma and the Riemann zetafunctions. By utilizing the explicit solution of this problem, we derive aplethora of novel identities involving the hypergeometric function.

The Laplace equation for the exterior of the Hankel contour and novel identities for hypergeometric functions