Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function for Laplace's equation in multiply connected circular domains are presented. The method is constructive and relies on the use of a special function known as the Schottky–Klein prime function associated with multiply connected circular domains. It is shown that all the important functions of potential theory, including the first-type Green's function, the modified Green's functions and the harmonic measures of a domain, can be written in terms of this prime function. A broad range of representative examples are given to demonstrate the efficacy of the method as well as a quantitative comparison, in special cases, with the results obtained using other independent methods.