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• Foundations of Data Science 数据科学基础

## 数学代写|复分析作业代写Complex function代考|Holomorphically Simply Connected Domains

The derivative-maximizing holomorphic function from an open set $U$ to the unit disc (as provided by Proposition 6.5.7) turns out to have some interesting special properties when $U$ resembles the disc in a certain sense. The type of resemblance that we wish to consider is called ‘holomorphic simple connectivity’, a concept that we introduced in Section 4.5. As noted there, this terminology is a temporary one that we use for convenience; it is not used universally (in particular, it is not found in most other texts on complex function theory).

The concept of ‘holomorphically simply connected’ (h.s.c.) arises naturally in our present context, as it did in Chapter 4 . However, it turns out to be implied by a more easily verified topological condition that is called simple connectivity. Simple connectivity is in fact a universally used mathematical concept; we shall discuss it in detail in Chapter 11 and demonstrate there that simple connectivity implies holomorphic simple connectivity. Recall the definition of holomorphic simple connectivity:

Definition 6.6.1. A connected open set $U \subseteq \mathbb{C}$ is holomorphically simply connected if, for each holomorphic function $f: U \rightarrow \mathbb{C}$, there is a holomorphic antiderivative $F$-that is, a function satisfying $F^{\prime}(z)=f(z)$.

EXAMPLE 6.6.2. As established in Chapter 4, open discs and open rectangles are holomorphically simply connected.

If $U_{1} \subseteq U_{2} \subseteq \ldots$ are holomorphically simply connected sets, then their union $U=\bigcup U_{j}$ is also holomorphically simply connected (Chapter 1 , Exercise 56). In particular, the plane $\mathbb{C}$ is holomorphically simply connected.
Theorem 6.6.3 (Riemann mapping theorem: analytic form). If $U$ is a holomorphically simply connected open set in $\mathbb{C}$ and $U \neq \mathbb{C}$, then $U$ is conformally equivalent to the unit disc.

## 数学代写|复分析作业代写Complex function代考|The Proof of the Analytic Form of the Riemann Mapping Theorem

Let $U$ be a holomorphically simply connected open set in $\mathbb{C}$ that is not equal to all of $\mathbb{C}$. Fix a point $P \in U$ and set
$\mathcal{F}={f: f$ is holomorphic on $U, f: U \rightarrow D,$,
$f$ is one-to-one, $f(P)=0}$.
We shall prove the following three assertions:
(1) $\mathcal{F}$ is nonempty.
(2) There is a function $f_{0} \in \mathcal{F}$ such that
$$\left|f_{0}^{\prime}(P)\right|=\sup {h \in \mathcal{F}}\left|h^{\prime}(P)\right| .$$ (3) If $g$ is any element of $\mathcal{F}$ such that $\left|g^{\prime}(P)\right|=\sup {h \in \mathcal{F}}\left|h^{\prime}(P)\right|$, then $g$ maps $U$ onto the unit $\operatorname{disc} D$.

The proof of assertion (1) is by direct construction. Statement (2) is almost the same as Proposition 6.5.7 (however there is now the extra element that the derivative-maximizing map must be shown to be one-toone). Statement (3) is the least obvious and will require some work: If the conclusion of (3) is assumed to be false, then we are able to construct an element $\hat{g} \in \mathcal{F}$ such that $\left|\hat{g}^{\prime}(P)\right|>\left|g^{\prime}(P)\right|$. Now we turn to the proofs.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Holomorphically Simply Connected Domains

“全纯简单连接” (hsc) 的概念在我们当前的上下文中很自然地出现，就像它在第 4 章中所做的那样。然而，事实证明，它被称为简单连通性的更容易验证的拓扑条件 所暗示。简单连通性实际上是一个普惼使用的数学概念；我们将在第 11 章洋细讨论它，并在那里证明简单连通性意味着全纯简单连通性。回想一下全纯简单连通性 的定义:

## 数学代写|复分析作业代写Complex function代考|The Proof of the Analytic Form of the Riemann Mapping Theorem

$\mathcal{F}=f: f \$ i$sholomorphicon$\$U, f: U \rightarrow D, \$, \$f \$ i$sone$-$to – one,$\$f(P)=0$.

(1) $\mathcal{F}$ 是非空的。
(2) 有一个功能 $f_{0} \in \mathcal{F}$ 这样
$$\left|f_{0}^{\prime}(P)\right|=\sup h \in \mathcal{F}\left|h^{\prime}(P)\right| .$$
(3) 如果 $g$ 是任何元素 $\mathcal{F}$ 这样 $\left|g^{\prime}(P)\right|=\sup h \in \mathcal{F}\left|h^{\prime}(P)\right|$ ， 然后 $g$ 地图 $U$ 到单位disc $D$.

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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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