﻿ 非均匀复变函数的级数理论 Series Theory of Heterogeneous Complex Variable Function

Vol. 09  No. 02 ( 2020 ), Article ID: 34223 , 9 pages
10.12677/AAM.2020.92021

Series Theory of Heterogeneous Complex Variable Function

Yanhui Yang1, Yujie Jiang1, Jicheng Tao2

1Department of Information and Computer Sciences, China Jiliang University, Hangzhou Zhejiang

2Department of Mathematics, China Jiliang University, Hangzhou Zhejiang

Received: Jan. 30th, 2020; accepted: Feb. 12th, 2020; published: Feb. 19th, 2020

ABSTRACT

In this paper, the series theory of complex variable function is applied to heterogeneous complex variable function. The convergence region and the radius of convergence ellipse of heterogeneous power series are established by using the theory of heterogeneous analytic function. At the same time, the Taylor’s theorem of heterogeneous analytic function is established by using the heterogeneous Cauchy integral formula and the high order derivative formula of Cauchy integral.

Keywords:Heterogeneous Power Series, Radius of Convergence Ellipse, Taylor’s Theorem of Heterogeneous Analytic Function

1中国计量大学信息科学系，浙江 杭州

2中国计量大学应用数学系，浙江 杭州

1. 引言

2. 预备知识

2.1. 非均匀复数的定义

${C}_{K}$ 中引入数乘

$z=a+jb,z\in {C}_{K},m\in R,mz=ma+jmb,$

${C}_{K}$ 中引入加法

${z}_{1}={a}_{1}+j{b}_{1},{z}_{2}={a}_{2}+j{b}_{2},{z}_{1}\in {C}_{K},{z}_{2}\in {C}_{K},$

${z}_{1}+{z}_{2}=\left({a}_{1}+{a}_{2}\right)+j\left({b}_{1}+{b}_{2}\right).$

2.2. 非均匀复变函数的定义及性质

$\frac{\Delta w}{\Delta z}=\frac{f\left(z\right)-f\left({z}_{0}\right)}{z-{z}_{0}}=\frac{f\left({z}_{0}+\Delta z\right)-f\left({z}_{0}\right)}{\Delta z}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(\Delta z\ne 0\right)$

${f}^{\prime }\left({z}_{0}\right)=\underset{\Delta z\to 0}{\mathrm{lim}}\frac{\Delta w}{\Delta z}=\underset{\Delta z\to 0}{\mathrm{lim}}\frac{f\left(z\right)-f\left({z}_{0}\right)}{z-{z}_{0}},$

1) 偏微分 ${p}_{x},{p}_{y},{q}_{x},{q}_{y}$ 在区域D内连续；

2) $p\left(x,y\right),q\left(x,y\right)$ 在区域D内满足非均匀C.-R.方程。

2.3. 非均匀复变函数的积分

$z=z\left(t\right)\text{}\left(\alpha \le t\le \beta \right)$

$a=z\left(\alpha \right)$ 为起点， $b=z\left(\beta \right)$ 为终点， $f\left(z\right)$ 沿C有定义，顺着C从a到b的方向在C上取分点： $a={z}_{0},{z}_{1},\cdots ,{z}_{n-1},{z}_{n}=b$，这样可以将曲线C划分为n个弧段，在从 ${z}_{t-1}$${z}_{t}$ 的每一个弧段上任取一点 ${\xi }_{t}$，那么

${S}_{n}=\underset{t=1}{\overset{n}{\sum }}f\left({\xi }_{t}\right)\Delta {z}_{t},\text{}\Delta {z}_{t}={z}_{t}-{z}_{t-1}$

${\int }_{C}f\left(z\right)\text{d}z={\int }_{C}u\text{d}x-kv\text{d}y+j{\int }_{C}v\text{d}x+u\text{d}y.$

3. 非均匀复级数理论文

3.1. 非均匀复数项级数基本性质

(3.1)

3.2. 一致非均匀收敛的非均匀复函数项级数

(3.2)

2)

2) 利用高阶导数的非均匀Cauchy积分公式(文献 [5] 定理4.8)和等式，结论是显然的。

3.3. 非均匀幂级数敛散性

(3.3)

(3.4)

(3.5)

。 (3.6)

1)的收敛椭圆半径为1，且，当，与复变函数的理论一致，且收敛到解析函数，当，转化成复数项级数的形式，利用复变函数级数理论，非均匀幂级数内绝对且内闭一致收敛于非均匀解析函数

2)的收敛半径为，且如1)一样的方法可知：非均匀幂级数在收敛椭圆域内绝对且内闭一致收敛于非均匀解析函数

3)的收敛半径为0，因此除处处发散。

3.4. 非均匀解析函数的泰勒级数展开

Series Theory of Heterogeneous Complex Variable Function[J]. 应用数学进展, 2020, 09(02): 178-186. https://doi.org/10.12677/AAM.2020.92021

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