A special case of Dirichlet's problem for two dimensions
J. C. Kluyver · Acta Mathematica · 1897
In a posthumous paper 1 of Riemann some indications are given about the construction of a real harmonic function W ill a plane with several circular holes, the function W taking assigned real values on each of the circular rims.Riemann's treatment of the problem is based on the theory of conformal representation.The given area is to be represented conformally on part of a Riemann surface, bounded by rectilinear rims, and then the desired function W can be readily found by means of an appropriate integral of the third kind.In 1877 the conformal representation of the plane with the holes was discussed anew by Schottky :, who arrived at a solution, depending on certain transcendental functions, not altered by the linear substitutions of a special discontinuous group, that was afterwards called by Poincar6 3 the symmetrical Kleinian group of the third family.In a second memoir Schottky 4 returned to this class of Kleinian functions and gave a full and ample discussion of their properties.By their means it is possible to treat in a direct way, and without having recourse to a previous mapping, the original Dirichlet's problem for the perforated plane.1 Gleichgewicht der Eleetricit~t auf Cylindern mit kreisf6rmigem Quersvhnitt und