In a preceding paper [17], these Fredholm eigen functions were applied to problems of conformal mapping of simply-connected plane domains.nIn the present paper, the Fredholm eigen value problem is studied in the case of multiply-connected plane domains. Various new difficulties arise in this case. The complementary region of a multiply-connected domain is a domain set and the number of trivial solutions of the problem with the eigen values increases. This fact necessitates a brief restatement of the basic definitions and concepts in Section 1.nIn Section 2, the dielectric Green’s functions gε(z, ) of a multiply-connected domain are discussed and their Fourier development in terms of the Fredholm eigen functions is given.nThe role of the one-parameter family gε(z, ) becomes particularly interesting when one studies the limit cases ε=0 and ε=00. This is done in Section 3.
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