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

Abstract

By using 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 identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation, which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundaryvalue problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis. © 2013 The Author(s) Published by the Royal Society. All rights reserved.