# 数学代写|复分析作业代写Complex function代考|Three Illustrative Applications of Inversion

#### Doug I. Jones

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## 数学代写|复分析作业代写Complex function代考|A Problem on Touching Circles

For our first problem, consider [3.14], in which we imagine that we are given two circles $A$ and $B$ that touch at $q$. As illustrated, we now construct the circle $C_0$ that touches $A$ and $B$ and whose centre lies on the horizontal line $L$ through the centres of $A$ and $B$. Finally, we construct the chain of circles $C_1, C_2$, etc., such that $C_{n+1}$ touches $C_n, A$, and $B$.

• The points of contact of the chain $\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2$, etc., all lie on a circle [dashed] touching $A$ and $B$ at $q$.
• If the radius of $C_n$ is $r_n$, then the height above $L$ of the centre of $C_n$ is $2 n r_n$. The figure illustrates this for $\mathrm{C}_3$.

Before reading further, see if you can prove either of these results using conventional geometric methods.

Inversion allows us to demonstrate both these results in a single elegant swoop. In [3.14], we have drawn the unique circle $\mathrm{K}$ centred at $q$ that cuts $\mathrm{C}_3$ at right angles. Thus inversion in $\mathrm{K}$ will map $C_3$ to itself, and it will map $A$ and $B$ to parallel vertical lines; see [3.15]. Check for yourself that the stated results are immediate consequences of this figure.

## 数学代写|复分析作业代写Complex function代考|The Point at Infinity

In discussing inversion we saw that results about lines could always be understood as special limiting cases of results about circles, simply by letting the radius tend to infinity. This limiting process is nevertheless tiresome and clumsy; how much better it would be if lines could literally be described as circles of infinite radius.
Here is another, related inconvenience. Inversion in the unit circle is a one-toone mapping of the plane to itself that swaps pairs of points. The same is true of the mapping $z \mapsto(1 / z)$. However, there are exceptions: no image point is presently associated with $z=0$, nor is 0 to be found among the image points.

To resolve both these difficulties, note that as $z$ moves further and further away from the origin, $(1 / z)$ moves closer and closer to 0 . Thus as $z$ travels to ever greater distances (in any direction), it is as though it were approaching a single point at infinity, written $\infty$, whose image is 0 . Thus, by definition, this point $\infty$ satisfies the following equations:
$$\frac{1}{\infty}=0, \quad \frac{1}{0}=\infty .$$
The addition of this single point at infinity turns the complex plane into the socalled extended complex plane. Thus we may now say, without qualification, that $z \mapsto(1 / z)$ is a one-to-one mapping of the extended plane to itself.

If a curve passes through $z=0$ then (by definition) the image curve under $z \mapsto(1 / z)$ will be a curve through the point at infinity. Conversely, if the image curve passes through 0 then the original curve passed through the point $\infty$. Since $z \mapsto(1 / z)$ swaps a circle through 0 with a line, we may now say that a line is just a circle that happens to pass through the point at infinity, and (without further qualification) inversion in a “circle” sends “circles” to “circles”.

This is all very tidy, but it leaves one feeling none the wiser. We are accustomed to using the symbol $\infty$ only in conjunction with a limiting process, not as a thing in its own right; how are we to grasp its new meaning as a definite point that is infinitely far away?

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|A Problem on Touching Circles

• 链条的接触点 $\mathrm{C}_0, \mathrm{C}_1, \mathrm{C}_2$ ，等等, 都躺在一个圆圈 上 [虚线] 感人 $A$ 和 $B$ 在 $q$.
• 如果半径为 $C_n$ 是 $r_n$ ，那么上面的高度 $L$ 的中心 $C_n$ 是 $2 n r_n$. 该图说明了这一点 $\mathrm{C}_3$.
在进一步阅读之前，看看您是否可以使用传统的几何方 法证明这些结果中的任何一个。
反转使我们能够一次优雅地展示这两个结果。在[3.14] 中，我们画出了独特的圆K集中于 $q$ 那削減 $\mathrm{C}_3$ 在直角。 因此反转 $\mathrm{K}$ 将映射 $C_3$ 到它自己，它会映射 $A$ 和 $B$ 平行垂 直线; 见[3.15]。自己检查一下，声明的结果是这个数字的直接结果。

## 数学代写|复分析作业代写Complex function代考|The Point at Infinity

$$\frac{1}{\infty}=0, \quad \frac{1}{0}=\infty$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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