Neumann Functions for Laplace's Equation for a Circular Cylinder of Finite Length

Abstract

In this paper general solutions for the Neumann problem for a circular cylinder of finite length are presented in terms of re-defined Neumann functions. The Neumann function, which has a zero normal derivative on the boundary, does not exist in a finite region according to Gauss's theorem. However, there exists a potential function due to a unit source which has a non-zero normal derivative. This function is defined as the Neumann function in the finite region. The Neumann functions for a circular cylinder of finite length are presented in three different forms. They correspond to three types of expressions of the potential function, called z-, r-, and ϕ-forms after Dougall's work. These Neumann functions play the same role in the Neumann type boundary value problem as Green's function in the Dirichlet type problem. The different forms are useful because they have their own regions of rapid convergence.