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# 数学代写|复分析代写Complex analysis代考|MATH3979 Topology on the fields of real and complex numbers

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## 数学代写|复分析代写Complex analysis代考|Topology on the fields of real and complex numbers

When reading this section, absorb the following main points of topology: open ball, open set, limit point, closed set. The main object of this section is to introduce limit points of sets so that we can in subsequent chapters talk about limits of functions, sequences, and series.

By a topology on a set we mean that some sets are declared open, subject to the conditions that the empty set and the whole set have to be open, that arbitrary unions of open sets be open, and that finite intersections of open sets be open. In any topology, the complement of an open set is called closed, but a set may be neither open nor closed. A topology can be imposed on any set, not just $\mathbb{R}$ or $\mathbb{C}$, but we focus on these two cases, and in fact we work only with the “standard”, or “Euclidean” topology.

## 数学代写|复分析代写Complex analysis代考|The Heine-Borel theorem

Closed and bounded sets in $\mathbb{C}$ and $\mathbb{R}$ have many excellent properties – we will for example see in Section $5.3$ that when a good (say continuous) real-valued function has a closed and bounded domain, then that function achieves a maximum and minimum value, et cetera. The concept of uniform continuity (introduced in Section 5.5) needs the fairly technical Heine-Borel theorems proved in this section.

Construction 3.6.1. (Halving closed and bounded subsets of $\mathbb{R}$ and quartering closed and bounded subsets of $\mathbb{C}$ ) Let $A$ be a bounded subset of of $\mathbb{R}$ or of $\mathbb{C}$, and let $P$ be a property that applies to some subsets of $A$. Boundedness of $A$ guarantees that $A$ fits inside a closed bounded rectangle $R_0$ of the form $\left(a_0, b_0\right) \times\left(c_0, d_0\right)$ in $\mathbb{C}$, with $c_0=0=d_0$ if $A$ is a subset of $\mathbb{R}$. The rectangle can be halved lengthwise and crosswise to get four equal closed subrectangles. In the next iteration we pick, if possible, one of these four closed quarter subrectangles such that its intersection with $A$ has property $P$. We call this subrectangle $R_1$. If $A$ is a subset of $\mathbb{R}$, then the length of $R_1$ is half the length of $R_0$, and otherwise the area of $R_1$ is a quarter of the area of $R_0$. In general, once we have $R_n$, we similarly pick a subrectangle $R_{n+1}$ such that $R_{n+1} \cap A$ has property $P$ and such that the sides of $R_{n+1}$ are half the lengths of the sides in $R_n$. Write $R_n=\left[a_n, b_n\right] \times\left[c_n, d_n\right]$ for some real numbers $a_n \leq b_n$ and $c_n \leq d_n$. By construction, for all $n, b_n-a_n=\left(b_0-a_0\right) / 2^n$, and
$$a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq \cdots \leq b_n \cdots \leq b_2 \leq b_1 \leq b_0 .$$
This means that $\left{a_1, a_2, a_3, \ldots\right}$ is a non-empty subset of $\mathbb{R}$ that is bounded above, so that by the Least upper bound property (Axiom 2.10.1), $a=\sup \left{a_1, a_2, a_3, \ldots\right}$ is a real number. Similarly, $b=\inf \left{b_1, b_2, b_3, \ldots\right}$ is a real number. Since $a \leq b_1, b_2, b_3, \ldots$, it follows that $a \leq b$. Suppose that $a<b$. Then by Exercise 2.10.6, there exists a positive integer $N$ such that $1 / 2^N<(b-a) /\left(b_0-a_0\right)$. But $a_N \leq a \leq b \leq b_N$, so that $0 \leq b-a \leq b_N-a_N=$ $\left(b_0-a_0\right) / 2^N<b-a$, which contradicts trichotomy. Thus $a=b$, i.e., we just proved that
$$\sup \left{a_1, a_2, a_3, \ldots\right}=\inf \left{b_1, b_2, b_3, \ldots\right}$$

## 数学代写|复分析代写Complex analysis代考|The Heine-Borel theorem

$$a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq \cdots \leq b_n \cdots \leq b_2 \leq b_1 \leq b_0$$

$1 / 2^N<(b-a) /\left(b_0-a_0\right)$. 但 $a_N \leq a \leq b \leq b_N$ ，以便 $0 \leq b-a \leq b_N-a_N=\left(b_0-a_0\right) / 2^N<b-a$ ，这与三分法 相矛盾。因此 $a=b$ ，即，我们刚刚证明了
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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。