Green's function for Laplace's equation in an infinite cylindrical cell

Abstract

The Green's function for Laplace's equation in an infinite-length cylinder with a homogeneous mixed boundary condition is considered. Its eigenfunction expansion converges slowly when the axial separation between the source and observation points is small compared to the cylinder radius, and diverges when the axial separation is zero. Applying a modified form of a contour integral method of Watson to an integral representation of the Green's function, a more general expansion of the Green's function is derived. Watson's original method had previously been applied to the case when the source and observation points were both on the axis of the cylinder. The expansion contains a free parameter which may be adjusted to give rapid convergence for any axial separation. It fails, however, when the source and observation points are both near the surface of the cylinder. For two special values of the parameter, the general expansion reduces to the eigenfunction expansion or to the integral representation. The derivation is somewhat obscure, but the resulting formula has a simple interpretation as the superposition of the potential of two related boundary value problems in finite-length cylinders. Some numerical results are given in the spatial region which previously could not be calculated, for a boundary condition approaching a homogeneous Neumann condition, and for a homogeneous Dirichlet condition.