Geodesic
The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 45-56

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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.
Keywords: generalized bi-axially symmetric Helmholtz equation; Green’s formula; fundamental solution; fourth double-layer potential; Appell’s hypergeometric functions of two variables; integral equations with double-layer potential density. 
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     author = {T. G. Ehrgashev},
     title = {The fourth double-layer potential for a generalized bi-axially symmetric {Helmholtz} equation},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {45--56},
     publisher = {mathdoc},
     number = {50},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2017_50_a3/}
}
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T. G. Ehrgashev. The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 45-56. http://geodesic.mathdoc.fr/item/VTGU_2017_50_a3/