Computability of series representations for Green's functions in a rectangle

Different series forms of Green's functions are analyzed for various boundary value problems stated for Laplace and Klein–Gordon equation in a rectangular region. The classical double-series representation for the Dirichlet problem for Laplace equation is converted into a single-series form, and a computational experiment is conducted to compare practical convergence of the two forms. By a partial summation of the single-series representation, the singular component of the Green's function is expressed in analytic form radically accelerating convergence of the remaining series for the regular component. Readily computable series forms are obtained for Green's functions of some mixed boundary value problems.