The Green's function of the Poisson equation on the non-concentric annular region

•Green's function of Poisson's equation for the non-concentric cylinder•Green's function of Poisson's equation for two cylinders of arbitrary radii and separation•Green's function are given in terms of elementary functions as well as in closed-form

In this article the two-dimensional Poisson equation is considered in the region between two non-concentric circular cylinders. Upon introducing bipolar coordinates the corresponding Green's function is found in form of a simple and rapidly converging series which can be formally summarized as a closed-form. Based on this result we additionally provide the Green's function for the conducting cylinder which is oriented parallel to a ground plane as well as for the case of two conducting cylinders.