On the solution of an integral equation arising in potential problems for circular and elliptic disks

This paper deals with a two-dimensional Fredholm integral equation of the first kind over the circular disk $S$, with a kernel of the form $g( \theta )/| {\bf r} - {\bf r}' | ,{\bf r} - {\bf r}' \in S$, where B is the angle between ${\bf r} - {\bf r}'$ and some reference direction. By expansions in Fourier series and in series involving Legendre functions, and by use of a new closed-form result for a Legendre-function integral, the integral equation is reduced to a system of linear equations for the expansion coefficients. It is shown that the system has a unique solution because of the Toeplitz structure of the system matrix. As an application, the electrostatic potential problem for a charged elliptic disk is discussed