Mathematical Methods in the Applied Sciences vol:33 issue:4 pages:431-438
In this paper we consider inhomogeneous generalized Helmholtz type equations $(\Delta + \lambda^2) u = f$ in the annulus of an arbitrary ball in $\R^3$ with given boundary conditions, where we assume that $\lambda$ is an arbitrary complex number. Applying the hypercomplex operator calculus, one can express the solutions in terms of quaternionic integral operators. The quantitative entities to be determined in order to calculate these integral operators in practice are the Cauchy kernel and the Bergman kernel for eigensolutions to the Dirac operator in $\R^3$. In contrast to the Cauchy kernel which is universal for all domains in $\R^3$, the Bergman kernel however depends on the domain. In this paper we give an explicit formula for the Bergman kernel of the annulus of a ball in $\R^3$ with arbitrary radii $0 < R_1 < R_2 < +\infty$ in terms of explicit special functions. With the
knowledge of the Bergman kernel the appearing integral operators can be evaluated fully analytically and thus provide us with explicit formulas for the solutions.