﻿ 多角形区域共形映射及其应用 The Conformal Mapping of Polygonal Domain and Its Application

Pure Mathematics
Vol.05 No.06(2015), Article ID:16437,7 pages
10.12677/PM.2015.56041

The Conformal Mapping of Polygonal Domain and Its Application

Huijie Ji

College of Mathematics and Computer Science, Shanxi Normal University, Linfen Shanxi

Received: Nov. 4th, 2015; accepted: Nov. 22nd, 2015; published: Nov. 27th, 2015

ABSTRACT

The conformal mapping is the important part of Function of Complex Variables and it has been widely used in various areas of science and technology. Firstly, this paper presents the conformal mapping of the upper half-plane onto polygonal domain and gives the Christoffel-Schwarz transform and its generalizing forms. And then we give the examples of the conformal mapping of the upper half-plane onto polygonal domain.

Keywords:The Conformal Mapping, Polygonal Domain, Christoffel-Schwarz Transform

1. 引言

2. 克里斯托费尔–施瓦茨变换

2.1. 共形映射的定义 [1]

(1) 伸缩率不变性：即在点附近，像点间的无穷小距离与原像点的无穷小距离之比的极限仅与有关，与过的曲线C之方向无关。

(2) 过的任意两曲线的夹角在变换下，既保持大小又保持方向。

2.2. 克里斯托费尔–施瓦茨变换

(2) 函数将上半平面共形映射成

(3)平面实轴上对应于平面多角形的顶点的那些点

(1)

(2)

3. 应用举例

，则对应的应为纯虚数。

(3)

Figure 1. The conformal mapping from polygonal domain of pipeline

Table 1. The data of conformal mapping from the polygonal domain of pipeline

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Figure 2. The conformal mapping from polygonal domain of upper half plane cutting by ray

Table 2. The data of the conformal mapping from polygonal domain of upper half plane cutting by ray

Figure 3. The conformal mapping from unit circle to square

Table 3. The data of conformal mapping from unit circle to square

(11)

(12)

(13)

(14)

(15)

(16)

4. 结论

The Conformal Mapping of Polygonal Domain and Its Application[J]. 理论数学, 2015, 05(06): 284-290. http://dx.doi.org/10.12677/PM.2015.56041

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