﻿ 模解析函数及其性质 Module Analytic Functions and Its Properties

Pure Mathematics
Vol.07 No.04(2017), Article ID:21474,7 pages
10.12677/PM.2017.74044

Module Analytic Functions and Its Properties

Chuanhua Jiang, Xiaohua Liang

Guangxi Normal University, Guilin Guangxi

Received: Jul. 6th, 2017; accepted: Jul. 20th, 2017; published: Jul. 25th, 2017

ABSTRACT

In this paper, the finite number is called the module derivative of complex function. And if exists module derivative at any point of some field, then is module analytic function over field. Let be a complex function, then we give a necessary condition, such that is a module analytic function as follows: which can be called module Cauchy-Riemann equation or shortly by M-C.R. equation. Furthermore, for module analytic function of field,we get the necessary and sufficient conditions: (1), satisfies the M-C.R. equation within the field. (2), satisfies the equation within the field. Finally, the correlations between module analytic function and several preexisting functions are discussed, including analysis function, semi-analytic function, and conjugate analytic function.

Keywords:Analysis Function, Cauchy-Riemann Equation, Module Analytic Function, Module Cauchy-Riemann Equation

1. 模解析函数

,

2. 主要定理及证明

(1)

,

,

(2)

(3)

, (4)

Figure 1. z + Δz tend to z)

(1) 偏导数在区域内存在；

(2)在区域内满足模C.-R.方程。

(1)在区域内满足模方程；

(2) 偏导数在区域内存在且

.

, , ,.

.

, , , ,

3. 模解析函数与其它函数类的关系

, , ,.

,

, , ,.

。根据半解析函数的定义，得不是半解析函数。

, , ,.

(1) 二元函数在区域内可微；

(2)在区域内满足共轭解析条件：

, , ,

Figure 2. The diagram of inclusion relation

Module Analytic Functions and Its Properties[J]. 理论数学, 2017, 07(04): 349-355. http://dx.doi.org/10.12677/PM.2017.74044

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