An application of Green's formula of a discrete function: Determination of periodicity moduli. II.

Introduction. Recently Opfer published a very interesting result [6] (also cf. [5]) in which he concerned himself with the problem of determining the modulus of a doubly connected domain by means of the difference method. In the present paper we shall consider a corresponding problem for a general multiply connected domain. It is known that for a non-degenerated N-ply connected domain (W^2) there exist N(N—ϊ)/2 quantities which are said to be periodicity moduli of the domain, which are conformally invariant, and which have an important meaning in the function theory. We shall concern ourselves with the problem of determining the system of periodicity moduli by means of the difference method (cf. Theorem 3.1 and Corollaries 2. 4, 3.1). Our method making effective use of Green's formula of a discrete function admits to deal with our problem by a unified principle. Also for a harmonic function u and a discrete harmonic function U on a domain G and a lattice R respectively which are constant on each boundary component of G and R, the monotonicity of the Dirichlet integral DG(u) and the summation SR(U) (see §2. 2) with respect to G and R is effectively utilized (cf. Lemmas 1.1, 2. 4, 2. 5 and 2. 6, and Theorem 2.1). For N=2 our main results (Theorem 3.1 and Corollary 3.1) coincide to Opfer's (Satz 7 of [6]). However even such a special case our method is deferent from his and is more simplified.