The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation

Applying a method of complex analysis (based upon analytic functions), R. P. Gilbert in 1969 constructed an integral representation of solutions of the generalized bi-axially symmetric Helmholtz equation. Fundamental solutions of this equation were constructed recently. In fact, when the spectral parameter is zero, fundamental solutions of the generalized bi-axially symmetric Helmholtz equation can be expressed in terms of Appell’s hypergeometric function of two variables of the second kind. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation are known, and only for the first one the theory of potential was constructed. In this paper, we aim at constructing a theory of double-layer potentials corresponding to the fourth fundamental solution. Using some properties of Appell’s hypergeometric functions of two variables, we prove limiting theorems and derive integral equations containing double-layer potential densities in the kernel.