An application of Green's formula of a discrete function: Determination of periodicity moduli. II.
An application of Green's formula of a discrete function: Determination of periodicity moduli. II.
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离散函数格林公式的应用：周期性模量的确定。
DOI：
10.2996/kmj/1138846121
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发表时间：
1970
期刊：
影响因子：
--
通讯作者：
Hisao Mizumoto
中科院分区：
文献类型：
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作者：
Hisao Mizumoto
导论.最近，Opfer发表了一个非常有趣的结果[6]（也参见。[5])在其中，他关心自己的问题，确定模的一个双重连接域的手段，差异的方法。在本文中，我们将考虑一般多连通域的一个相应问题。已知对于非退化的N重连通域（W^2），存在N（N-n）/2个量，称之为该域的周期模，它们是共形不变的，在函数论中具有重要意义.我们将研究用差分法确定周期性模量系统的问题（参见第二章）。定理3.1和推论2。4、3.1）。我们的方法有效地利用了离散函数的绿色公式，使我们的问题得到了统一的处理。同样，对于区域G和格R上的调和函数u和离散调和函数U，它们在G和R的每个边界分量上都是常数，Dirichlet积分DG（u）和求和SR（U）的单调性（见§2. 2)关于G和R的有效利用（cf.引理1.1，2.四二5和2.（定理2.1）。对于N=2，我们的主要结果（定理3.1和推论3.1）与Opfer的结果（[6]的Satz 7）一致。然而，即使是这样一种特殊情况，我们的方法也不同于他的方法，并且更简化。
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.