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

TL;DR

This paper uses conformal mappings and the global relation to solve boundary value problems for the Laplace equation, leading to new identities for hypergeometric functions, especially related to the Hankel contour and special functions.

Contribution

It introduces a method to derive novel hypergeometric identities from boundary value problems involving the Hankel contour using explicit solutions and conformal mappings.

Findings

Derived new identities for hypergeometric functions.

Solved the Neumann problem for Laplace outside the Hankel contour.

Connected boundary value problems to special function identities.

Abstract

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.