## 数学代写|黎曼曲面代写Riemann surface代考|MAST90056

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

## 数学代写|黎曼曲面代写Riemann surface代考|Divisors and the Abel Theorem

In order to analyze functions and differentials on Riemann surfaces, one characterizes them in terms of their zeros and poles. It is convenient to consider formal sums of points on $\mathcal{R}$. (Later these points will become zeros and poles of functions and differentials).

Definition 24. A formal linear combination
$$D=\sum_{j=1}^N n_j P_j, \quad n_j \in \mathbb{Z}, P_j \in \mathcal{R}$$
is called a divisor on the Riemann surface $\mathcal{R}$. The sum
$$\operatorname{deg} D=\sum_{j=1}^N n_j$$
is called the degree of $D$.
The set of all divisors with the obviously defined group operations
$$n_1 P+n_2 P=\left(n_1+n_2\right) P, \quad-D=\sum_{j=1}^N\left(-n_j\right) P_j$$
forms an Abelian group $\operatorname{Div}(\mathcal{R})$. A divisor (1.69) with all $n_j \geq 0$ is called positive (or integral, or effective). This notion allows us to define a partial ordering in $\operatorname{Div}(\mathcal{R})$
$$D \leq D^{\prime} \Longleftrightarrow D^{\prime}-D \geq 0 .$$

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Roch Theorem

Let $D_{\infty}$ be a positive divisor on $\mathcal{R}$. A natural problem is to describe the vector space of meromorphic functions with poles at $D_{\infty}$ only. More generally, let $D$ be a divisor on $\mathcal{R}$. Let us consider the vector space
$$L(D)={f \text { meromorphic functions on } \mathcal{R} \mid(f) \geq-D \text { or } f \equiv 0} .$$
Let us split
$$-D=D_0-D_{\infty}$$
into negative and positive parts
$$D_0=\sum n_i P_i, \quad D_{\infty}=\sum m_k Q_k,$$
where both $D_0$ and $D_{\infty}$ are positive. The space $L(D)$ of dimension
$$l(D)=\operatorname{dim} L(D)$$
consists of the meromorphic functions with zeros of order at least $n_i$ at $P_i$ and with poles of order at most $m_k$ at $Q_k$.
Similarly, let us denote by
$$H(D)={\Omega \text { Abelian differential on } \mathcal{R} \mid(\Omega) \geq D \text { or } \Omega \equiv 0}$$
the corresponding vector space of differentials, and by
$$i(D)=\operatorname{dim} H(D)$$
its dimension, which is called the index of speciality of $D$.
It is easy to see that $l(D)$ and $i(D)$ depend only on the divisor class of $D$, and
$$i(D)=l(C-D),$$
where $C$ is the canonical divisor class. Indeed, let $\Omega_0$ be a non-zero Abelian differential and $C=\left(\Omega_0\right)$ be its divisor. The map $H(D) \rightarrow L(C-D)$ defined by
$$H(D) \ni \Omega \longrightarrow \frac{\Omega}{\Omega_0} \in L(C-D)$$
is an isomorphism of linear spaces, which implies $i(D)=l(C-D)$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Divisors and the Abel Theorem

$$D=\sum_{j=1}^N n_j P_j, \quad n_j \in \mathbb{Z}, P_j \in \mathcal{R}$$

$$\operatorname{deg} D=\sum_{j=1}^N n_j$$

$$n_1 P+n_2 P=\left(n_1+n_2\right) P, \quad-D=\sum_{j=1}^N\left(-n_j\right) P_j$$

$$D \leq D^{\prime} \Longleftrightarrow D^{\prime}-D \geq 0 .$$

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Roch Theorem

$$L(D)={f \text { meromorphic functions on } \mathcal{R} \mid(f) \geq-D \text { or } f \equiv 0} .$$

$$-D=D_0-D_{\infty}$$

$$D_0=\sum n_i P_i, \quad D_{\infty}=\sum m_k Q_k,$$

$$l(D)=\operatorname{dim} L(D)$$

$$H(D)={\Omega \text { Abelian differential on } \mathcal{R} \mid(\Omega) \geq D \text { or } \Omega \equiv 0}$$

$$i(D)=\operatorname{dim} H(D)$$

$$i(D)=l(C-D),$$

$$H(D) \ni \Omega \longrightarrow \frac{\Omega}{\Omega_0} \in L(C-D)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|MA5253

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

## 数学代写|黎曼曲面代写Riemann surface代考|Divisors and the Abel Theorem

In order to analyze functions and differentials on Riemann surfaces, one characterizes them in terms of their zeros and poles. It is convenient to consider formal sums of points on $\mathcal{R}$. (Later these points will become zeros and poles of functions and differentials).

Definition 24. A formal linear combination
$$D=\sum_{j=1}^N n_j P_j, \quad n_j \in \mathbb{Z}, P_j \in \mathcal{R}$$
is called a divisor on the Riemann surface $\mathcal{R}$. The sum
$$\operatorname{deg} D=\sum_{j=1}^N n_j$$
is called the degree of $D$.
The set of all divisors with the obviously defined group operations
$$n_1 P+n_2 P=\left(n_1+n_2\right) P, \quad-D=\sum_{j=1}^N\left(-n_j\right) P_j$$
forms an Abelian group $\operatorname{Div}(\mathcal{R})$. A divisor (1.69) with all $n_j \geq 0$ is called positive (or integral, or effective). This notion allows us to define a partial ordering in $\operatorname{Div}(\mathcal{R})$
$$D \leq D^{\prime} \Longleftrightarrow D^{\prime}-D \geq 0 .$$

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Roch Theorem

Let $D_{\infty}$ be a positive divisor on $\mathcal{R}$. A natural problem is to describe the vector space of meromorphic functions with poles at $D_{\infty}$ only. More generally, let $D$ be a divisor on $\mathcal{R}$. Let us consider the vector space
$$L(D)={f \text { meromorphic functions on } \mathcal{R} \mid(f) \geq-D \text { or } f \equiv 0} .$$
Let us split
$$-D=D_0-D_{\infty}$$
into negative and positive parts
$$D_0=\sum n_i P_i, \quad D_{\infty}=\sum m_k Q_k,$$
where both $D_0$ and $D_{\infty}$ are positive. The space $L(D)$ of dimension
$$l(D)=\operatorname{dim} L(D)$$
consists of the meromorphic functions with zeros of order at least $n_i$ at $P_i$ and with poles of order at most $m_k$ at $Q_k$.
Similarly, let us denote by
$$H(D)={\Omega \text { Abelian differential on } \mathcal{R} \mid(\Omega) \geq D \text { or } \Omega \equiv 0}$$
the corresponding vector space of differentials, and by
$$i(D)=\operatorname{dim} H(D)$$
its dimension, which is called the index of speciality of $D$.
It is easy to see that $l(D)$ and $i(D)$ depend only on the divisor class of $D$, and
$$i(D)=l(C-D),$$
where $C$ is the canonical divisor class. Indeed, let $\Omega_0$ be a non-zero Abelian differential and $C=\left(\Omega_0\right)$ be its divisor. The map $H(D) \rightarrow L(C-D)$ defined by
$$H(D) \ni \Omega \longrightarrow \frac{\Omega}{\Omega_0} \in L(C-D)$$
is an isomorphism of linear spaces, which implies $i(D)=l(C-D)$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Divisors and the Abel Theorem

$$D=\sum_{j=1}^N n_j P_j, \quad n_j \in \mathbb{Z}, P_j \in \mathcal{R}$$

$$\operatorname{deg} D=\sum_{j=1}^N n_j$$

$$n_1 P+n_2 P=\left(n_1+n_2\right) P, \quad-D=\sum_{j=1}^N\left(-n_j\right) P_j$$

$$D \leq D^{\prime} \Longleftrightarrow D^{\prime}-D \geq 0 .$$

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Roch Theorem

$$L(D)={f \text { meromorphic functions on } \mathcal{R} \mid(f) \geq-D \text { or } f \equiv 0} .$$

$$-D=D_0-D_{\infty}$$

$$D_0=\sum n_i P_i, \quad D_{\infty}=\sum m_k Q_k,$$

$$l(D)=\operatorname{dim} L(D)$$

$$H(D)={\Omega \text { Abelian differential on } \mathcal{R} \mid(\Omega) \geq D \text { or } \Omega \equiv 0}$$

$$i(D)=\operatorname{dim} H(D)$$

$$i(D)=l(C-D),$$

$$H(D) \ni \Omega \longrightarrow \frac{\Omega}{\Omega_0} \in L(C-D)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|MAT00111M

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

## 数学代写|黎曼曲面代写Riemann surface代考|Holomorphic Mappings

Definition 7. A mapping
$$f: M \rightarrow N$$
between Riemann surfaces is called holomorphic if for every local parameter $(U, z)$ on $M$ and every local parameter $(V, w)$ on $N$ with $U \cap f^{-1}(V) \neq \emptyset$, the mapping
$$w \circ f \circ z^{-1}: z\left(U \cap f^{-1}(V)\right) \rightarrow w(V)$$
is holomorphic.
A holomorphic mapping to $\mathbb{C}$ is called a holomorphic function, a holomorphic mapping to $\hat{\mathbb{C}}$ is called a meromorphic function.

The following lemma characterizes the local behavior of holomorphic mappings.

Lemma 1. Let $f: M \rightarrow N$ be a holomorphic mapping. Then for any $a \in M$ there exist $k \in \mathbb{N}$ and local parameters $(U, z),(V, w)$ such that $a \in U, f(a) \in V$ and $F=w \circ f \circ z^{-1}: z(U) \rightarrow w(V)$ equals
$$F(z)=z^k .$$
Corollary 1. Let $f: M \rightarrow N$ be a non-constant holomorphic mapping, then $f$ is open, i.e., the image of an open set is open.

If $M$ is compact then $f(M)$ is compact as a continuous image of a compact set and open due to the previous claim. This implies that in this case the corresponding non-constant holomorphic mapping is surjective and its image $N=f(M)$ compact.

We see that there exist no non-constant holomorphic mappings $f: M \rightarrow \mathbb{C}$, which is the issue of the classical Liouville theorem.

## 数学代写|黎曼曲面代写Riemann surface代考|Algebraic Curves as Coverings

Let $C$ be a non-singular algebraic curve (1.2) and $\hat{C}$ its compactification. The map
$$(\mu, \lambda) \rightarrow \lambda$$
is a holomorphic covering $\hat{C} \rightarrow \hat{\mathbb{C}}$. If $N$ is the degree of the polynomial $\mathcal{P}(\mu, \lambda)$ in $\mu$
$$\mathcal{P}(\mu, \lambda)=\mu^N p_N(\lambda)+\mu^{N-1} p_{N-1}(\lambda)+\ldots+p_0(\lambda),$$
where all $p_i(\lambda)$ are polynomials, then $\lambda: \hat{C} \rightarrow \hat{\mathbb{C}}$ is an $N$-sheeted covering, see Fig. 1.4.

The points with $\partial \mathcal{P} / \partial \mu=0$ are the branch points of the covering $\lambda$ : $C \rightarrow \mathbb{C}$. At these points $\partial \mathcal{P} / \partial \lambda \neq 0$, and $\mu$ is a local parameter. The derivative of $\lambda$ with respect to the local parameter vanishes
$$\frac{\partial \lambda}{\partial \mu}=-\frac{\partial \mathcal{P} / \partial \mu}{\partial \mathcal{P} / \partial \lambda}=0,$$
which characterizes (1.19) the branch points of the covering (1.21). In the same way the map $(\mu, \lambda) \mapsto \mu$ is a holomorphic covering of the $\mu$-plane. The branch points of this covering are the points with $\partial \mathcal{P} / \partial \lambda=0$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Holomorphic Mappings

$$f: M \rightarrow N$$

$$w \circ f \circ z^{-1}: z\left(U \cap f^{-1}(V)\right) \rightarrow w(V)$$

$$F(z)=z^k .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Algebraic Curves as Coverings

$$(\mu, \lambda) \rightarrow \lambda$$

$$\mathcal{P}(\mu, \lambda)=\mu^N p_N(\lambda)+\mu^{N-1} p_{N-1}(\lambda)+\ldots+p_0(\lambda),$$

$$\frac{\partial \lambda}{\partial \mu}=-\frac{\partial \mathcal{P} / \partial \mu}{\partial \mathcal{P} / \partial \lambda}=0,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic functions

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic functions

We now turn to the study of meromorphic functions on Riemann surfaces of genus 1 .
The only Riemann surface of genus 0 is the Riemann sphere $\mathbb{P}^1=\mathbb{C} \cup{\infty}$. (This is not obvious: we are saying that any abstract Riemann surface structure on the 2 -sphere ends up being isomorphic to the standard one. If you recall hat Riemann surface structures can be defined by gluing, you see why this is not a simple consequence of any definition). On $\mathbb{P}^1$, the meromorphic functions are rational, and those we understand quite explicitly; so it is natural to study tori next.

The tori we shall study are of the form $\mathbb{C} / L$, where $L \subset \mathbb{C}$ is a lattice – a free abelian subgroup for which the quotient is a topological torus. A less tautological definition is, viewing $\mathbb{C}$ as $\mathbb{R}^2$, that $L$ should be generated over $\mathbb{Z}$ by two vectors which are not parallel. Calling them $\omega_1$ and $\omega_2$, the conditions are
$$\omega_1, \omega_2 \neq 0 \quad \text { and } \quad \frac{\omega_1}{\omega_2} \notin \mathbb{R} \text {. }$$’

8.1 Exercise: Show, if $\omega_1 / \omega_2 \in \mathbb{R}$, that $\mathbb{Z} \omega_1+\mathbb{Z} \omega_2 \subset \mathbb{C}$ is either generated over $\mathbb{Z}$ by a single vector, or else its points are dense on a line. (The two cases correspond to $\omega_1 / \omega_2 \in \mathbb{Q}$ and $\omega_1 / \omega_2 \in \mathbb{R} \backslash \mathbb{Q}$.)

By definition, a function $f$ is holomorphic on an open subset $U \subseteq \mathbb{C} / L$ iff $f \circ \pi$ is holomorphic on $\pi^{-1}(U) \subseteq \mathbb{C}$, where $\pi: \mathbb{C} \rightarrow \mathbb{C} / L$ is the projection.
Note that a ‘fundamental domain’ for the action of $L$ on $\mathbb{C}$ is the ‘period parallelogram’ Strictly speaking, to represent each point only once, we should take the interior of the parallelogram, two open edges and a single vertex; but it is more sensible to view $\mathbb{C} / L$ as arising from the closed parallelogram by identifying opposite sides. The notion of holomorphicity is pictorially clear now, even at a boundary point $P$ – we require matching functions on the two halfneighbourhoods of $P$.

## 数学代写|黎曼曲面代写Riemann surface代考|Remark

8.2 Remark: Division by $\omega_1$ turns the period parallelogram into the form depicted in Fig. 8.2, with $\tau=\omega_2 / \omega_1 \notin \mathbb{R}$.

Another presentation of the Riemann surface $T=\mathbb{C} / L$ is then visibly as $\mathbb{C}^* / \mathbb{Z}$, where the abelian group $\mathbb{Z}$ is identified with the multiplicative subgroup of $\mathbb{C}^$ generated by $q=e^{\pi i \tau}$. We have a map $\exp : \mathbb{C} \rightarrow \mathbb{C}^$ which descends to an isomorphism of Riemann surfaces, between $\mathbb{C} / L$ and $\mathbb{C}^* /\left{q^{\mathbb{Z}}\right}$

Returning to the $\mathbb{C} / L$ description, we see that functions on $T$ correspond to doubly periodic functions on $\mathbb{C}$, that is, functions satisfying
$$f\left(u+\omega_1\right)=f\left(u+\omega_2\right)=f(u)$$
for all $u \in \mathbb{C}$. For starters, we note the following:
8.3 Proposition: Any doubly periodic holomorphic function on $\mathbb{C}$ is constant.
First Proof: Global holomorphic functions on $\mathbb{C} / L$ are constant.
Second Proof: By Liouville’s theorem, bounded holomorphic functions on $\mathbb{C}$ are constant.

8.4 Definition: An elliptic function is a doubly periodic meromorphic function on $\mathbb{C}$.
Elliptic functions are thus meromorphic functions on a torus $\mathbb{C} / L$. The reason for the name is lost in the dawn of time. (Really, elliptic functions can be used to express the arc-length on the ellipse.)

Constructing the first example of an elliptic function takes some work. We shall in fact describe them all; but we must start with some generalities.
8.5 Theorem: Let $z_1, \ldots, z_n$ and $p_1, \ldots, p_m$ denote the zeroes and poles of a non-constant elliptic function $f$ in the period parallelogram, repeated according to multiplicity. Then:
(i) $m=n$,
(ii) $\sum_{k=1}^m \operatorname{Res}{p_k}(f)=0$, (iii) $\sum{k=1}^n z_k=\sum_{k=1}^m p_k(\bmod L)$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic functions

$$\omega_1, \omega_2 \neq 0 \quad \text { and } \quad \frac{\omega_1}{\omega_2} \notin \mathbb{R} \text {. }$$”

8.1练习:如果$\omega_1 / \omega_2 \in \mathbb{R}$，表示$\mathbb{Z} \omega_1+\mathbb{Z} \omega_2 \subset \mathbb{C}$是由单个向量在$\mathbb{Z}$上生成的，否则它的点在一条线上是密集的。(这两种情况对应于$\omega_1 / \omega_2 \in \mathbb{Q}$和$\omega_1 / \omega_2 \in \mathbb{R} \backslash \mathbb{Q}$。)

## 数学代写|黎曼曲面代写Riemann surface代考|Remark

8.2注:除以$\omega_1$，周期平行四边形得到图8.2所示的形式，其中有$\tau=\omega_2 / \omega_1 \notin \mathbb{R}$。

$$f\left(u+\omega_1\right)=f\left(u+\omega_2\right)=f(u)$$

8.3命题:$\mathbb{C}$上的任何双周期全纯函数都是常数。

8.4定义:椭圆函数是$\mathbb{C}$上的双周期亚纯函数。

8.5定理:设$z_1, \ldots, z_n$和$p_1, \ldots, p_m$为周期平行四边形中一个非常椭圆函数$f$的零点和极点，按多重重复。然后:
(i) $m=n$;
(ii) $\sum_{k=1}^m \operatorname{Res}{p_k}(f)=0$， (iii) $\sum{k=1}^n z_k=\sum_{k=1}^m p_k(\bmod L)$。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|Unique Presentations of meromorphic functions on P1

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Unique Presentations of meromorphic functions on $\mathbb{P}^1$

We shall describe more closely the holomorphic maps from $\mathbb{P}^1$ to itself. By the results (3.14) of the previous lecture, these are the same as the meromorphic functions on $\mathbb{P}^1$, plus the constant $\operatorname{map} \infty$.

Recall that a rational function $R(z)$ is one expressible as a ratio of two polynomials, $p(z) / q(z)$ ( $q$ not identically zero). Clearly, it is meromorphic. We may assume $p$ and $q$ to have no common factors, in which case we call $\max (\operatorname{deg} p, \operatorname{deg} q)$ the degree of $R(z)$.
4.8 Theorem: Every meromorphic function on $\mathbb{P}^1$ is rational.
We shall prove two stronger statements.
4.9 Theorem (Unique Presentation by principal parts): A meromorphic function on $\mathbb{P}^1$ is uniquely expressible as
$$p(z)+\sum_{i, j} \frac{c_{i j}}{\left(z-p_i\right)^j}$$
where $p(z)$ is a polynomial, the $c_{i j}$ are constants and the sum is finite.
4.10 Remark: The $p_i$ are the finite poles of the function.
Proof: Recall that, near a pole $p$, a meromorphic function has a convergent Laurent expansion:
$$a_n(z-p)^{-n}+a_{-n+1}(z-p)^{-n+1}+\cdots+a_{-1}(z-p)^{-1}+\sum_{k \geq 0} a_k(z-p)^k$$
and the negative powers form the principal part of the series.

## 数学代写|黎曼曲面代写Riemann surface代考|Global consequences of the theorem on the local form

5.1 Theorem: Let $f: R \rightarrow S$ be a non-constant holomorphic map, with $R$ connected and compact. Then $f$ surjects onto a compact connected component of $S$.
5.2 Corollaries:
(i) A non-constant holomorphic map between compact connected Riemann surfaces is surjective.
(ii) A global holomorphic function on a compact Riemann surface is constant.
(iii) (Fundamental Theorem of Algebra) A non-constant complex polynomial has a least one root.

Proof of the theorem: $f$ is open and continuous and $R$ is compact, so $f(R)$ is open in $S$ and compact, hence closed. As $R$ is also connected, $f(R)$ is connected, so it is a connected component of $S .(S=f(R) \cup(S \backslash f(R))$ with $f(R)$ and $S \backslash f(R)$ both open.)
Proof of the corollaries:
(i) Clear from the theorem and connectedness of $S$.

(ii) A holomorphic function determines a map to $\mathbb{C}$, hence a holomorphic map to $\mathbb{P}^1$. By the previous corollary, the image of any non-constant map would be contain $\infty$; so the map must be constant.
(iii) A polynomial determines a holomorphic map $\mathbb{P}^1 \rightarrow \mathbb{P}^1$. If not constant, the image of this map must contain 0 , so the polynomial must have a root.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Unique Presentations of meromorphic functions on $\mathbb{P}^1$

4.8定理:$\mathbb{P}^1$上的每一个亚纯函数都是有理的。

4.9定理(主部唯一表示):$\mathbb{P}^1$上的亚纯函数可唯一表示为
$$p(z)+\sum_{i, j} \frac{c_{i j}}{\left(z-p_i\right)^j}$$

4.10注:$p_i$是函数的有限极点。

$$a_n(z-p)^{-n}+a_{-n+1}(z-p)^{-n+1}+\cdots+a_{-1}(z-p)^{-1}+\sum_{k \geq 0} a_k(z-p)^k$$

## 数学代写|黎曼曲面代写Riemann surface代考|Global consequences of the theorem on the local form

5.1定理:设$f: R \rightarrow S$为非常全纯映射，$R$连通且紧致。然后$f$投射到$S$的紧密连接组件上。
5.2推论:
(1)紧连通黎曼曲面间的非常全纯映射是满射的。
(ii)紧致Riemann曲面上的全局全纯函数是常数。
(3)(代数基本定理)一个非常复数多项式至少有一个根。

(i)由$S$的定理和连通性可知。

(ii)一个全纯函数决定了一个到$\mathbb{C}$的映射，因此一个到$\mathbb{P}^1$的全纯映射。根据前面的推论，任何非常数映射的图像都包含$\infty$;所以映射必须是常数。
(iii)多项式确定一个全纯映射$\mathbb{P}^1 \rightarrow \mathbb{P}^1$。如果不是常数，这个映射的像必须包含0，所以多项式必须有一个根。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|Concrete Riemann Surfaces

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Concrete Riemann Surfaces

Historically, Riemann surfaces arose as graphs of analytic functions, with multiple values, defined over domains in $\mathbb{C}$. Inspired by this, we now give a precise definition of a concrete Riemann surface; but we need a preliminary notion.

2.2 Definition: A complex function $F(z, w)$ defined in an open set in $\mathbb{C}^2$ is called holomorphic if, near each point $\left(z_0, w_0\right)$ in its domain, $F$ has a convergent power series expansion
$$F(z, w)=\sum_{m, n \geq 0} F_{m n}\left(z-z_0\right)^m\left(w-w_0\right)^n .$$
The basic properties of 2-variable power series are assigned to Problem 1.4; in particular, $F$ is differentiable in its region of convergence, and we can differentiate term by term.
2.3 Definition: A subset $S \subseteq \mathbb{C}^2$ is called a (concrete, possibly singular) Riemann surface if, for each point $s \in S$, there is a neighbourhood $U$ of $s$ and a holomorphic function $F$ on $U$ such that $S \cap U$ is the zero-set of $F$ in $U$; moreover, we require that $\partial^n F / \partial w^n(s) \neq 0$ for some $n$.
In particular, the continuity of $F$ implies that $S$ is locally closed. The condition $\partial^n F / \partial w^n(s) \neq 0$ rules out vertical lines through $s$, which cannot reasonably be viewed as ‘graphs’. (Indeed, we can see from the power series expansion that $S \cap U$ will contain a vertical line precisely when $F_{0 n}=0$ for all $\left.n.\right)$
2.4 Definition: The Riemann surface is called non-singular at $s \in S$ if a function $F$ can be found with the gradient vector $(\partial F / \partial z, \partial F / \partial w)$ non-zero at $s$.
2.5 Theorem (Local structure of non-singular Riemann surfaces):
(i) Assume $\partial F / \partial w(s) \neq 0$. Then, in some neighbourhood of $s, S$ is the graph of a holomorphic function $w=w(z)$.
(ii) Assume $\partial F / \partial z(s) \neq 0$. Then, in some neighbourhood of $s, S$ is the graph of a holomorphic function $z=z(w)$.
(iii) Assume both. Then, the two holomorphic functions above are inverse to each other.

## 数学代写|黎曼曲面代写Riemann surface代考|Abstract Riemann surfaces

For most of the course, we shall consider Riemann surfaces from an abstract point of view. This suffices to establish their general properties, and dispenses with unnecessary embedding information. (Moreover, smoothness is built in, whereas in the embedded case it must be checked). However, the abstract definition is somewhat complicated and less intuitive. One way to motivate their introduction is by the following observation.
2.7 Proposition: Every Riemann surface in $\mathbb{C}^2$ is non-compact. (Proof in the next lecture).
This is clear for a Riemann surface defined as the zero-set of an algebraic equation $P(z, w(z))=0$; it projects surjectively to the complex z-plane, Because the image of a compact set under a continuous map is compact, it follows that the solution-set is not compact.

So, there is an obstacle to constructing compact Riemann surfaces, such as the torus without punctures, as graphs of multi-valued functions within $\mathbb{C}^2$. On the other hand, it’s easy to produce compact topological surfaces with enough analytic structure to be worthy of Riemann’s name. Here are two examples:
(2.8) The Riemann sphere $\mathbb{C} \cup{\infty}=\mathbb{P}^1$ (Fig. 2.2).
The topological description of how $\mathbb{C} \cup{\infty}$ becomes a sphere is best illustrated by the stereographic projection, in which points going off to $\infty$ in the plane converge to the north pole in the sphere. (The south pole maps to 0.)

We can understand $\mathbb{P}^1$ as a Riemann surface is by regarding $z^{-1}=w$ as a local coordinate near $\infty$. We say that a function $f$ defined in the neighbourhood of $\infty$ on $\mathbb{P}^1$ is holomorphic if the following function is holomorphic, in a neighbourhood of $w=0$ :
$$w \mapsto \begin{cases}f\left(w^{-1}\right), & \text { if } w \neq 0 \ f(\infty), & \text { if } w=0\end{cases}$$
There is another descrition of $\mathbb{P}^1$ as a Riemann surface. Consider two copies of $\mathbb{C}$, with coordinates $z$ and $w$. The map $w=z^{-1}$ identifies $\mathbb{C} \backslash{0}$ in the $z$-plane with $\mathbb{C} \backslash{0}$ in the $w$-plane, in analytic and invertible fashion. (We say that the map $z \mapsto w=z^{-1}$ from $\mathbb{C}^$ to $\mathbb{C}^$ is bianalytic, or biholomorphic.) Define a new topological space by gluing the two copies of $\mathbb{C}$ along this identification. Clearly, we get a topological sphere, but now there is an obvious notion of holomorphic function on it: we have $\mathbb{P}^1=\mathbb{C}{(z)} \cup \mathbb{C}{(w)}$, and we declare a function $f$ on $\mathbb{P}^1$ to be homomorphic precisely if its restrictions to the open sets $\mathbb{C}=\mathbb{P}^1 \backslash{\infty}$ and $\mathbb{C}=\mathbb{P}^1 \backslash{0}$ are holomorphic.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Concrete Riemann Surfaces

2.2定义:定义在$\mathbb{C}^2$中的开集中的复函数$F(z, w)$，如果在其定义域内的每个点$\left(z_0, w_0\right)$附近，$F$有收敛幂级数展开，则称为全纯的
$$F(z, w)=\sum_{m, n \geq 0} F_{m n}\left(z-z_0\right)^m\left(w-w_0\right)^n .$$

2.3定义:如果对于每个点$s \in S$，存在$s$的邻域$U$和$U$上的全纯函数$F$，使得$S \cap U$是$U$中的$F$的零集，则子集$S \subseteq \mathbb{C}^2$被称为(具体的，可能是奇异的)Riemann曲面;此外，对于一些$n$，我们需要$\partial^n F / \partial w^n(s) \neq 0$。

2.4定义:如果函数$F$在$s$处梯度向量$(\partial F / \partial z, \partial F / \partial w)$不为零，则称为在$s \in S$处的黎曼曲面非奇异。
2.5定理(非奇异黎曼曲面的局部结构):

(三)两者都假定。那么，上述两个全纯函数互为逆。

## 数学代写|黎曼曲面代写Riemann surface代考|Abstract Riemann surfaces

2.7命题:$\mathbb{C}^2$中的每一个黎曼曲面都是非紧的。(下节课再证明)

(2.8)黎曼球$\mathbb{C} \cup{\infty}=\mathbb{P}^1$(图2.2)。

$$w \mapsto \begin{cases}f\left(w^{-1}\right), & \text { if } w \neq 0 \ f(\infty), & \text { if } w=0\end{cases}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups

We can now officially define the main object of study of this chapter, one-parameter semigroups on Riemann surfaces.

Definition 5.2.1. Let $X$ be a Riemann surface. A one-parameter semigroup of holomorphic maps (briefly, a one-parameter semigroup) on $X$ is a continuous semigroup homomorphism $\Phi$ from $\mathbb{R}^{+}$to $\operatorname{Hol}(X, X)$ endowed with the composition. A one-parameter group of holomorphic maps on $X$ is a continuous group homomorphism from $(\mathbb{R},+)$ to $\operatorname{Hol}(X, X)$. When $t \in \mathbb{R}^{+}$and $z \in X$, we shall often write $\Phi_t(z)$ or $\Phi(t, z)$ instead of $\Phi(t)(z)$. The trivial one-parameter semigroup is the trivial homomorphism $\Phi_t \equiv \operatorname{id}_X$ for all $t \in \mathbb{R}^{+}$. Finally, we shall say that a nontrivial one-parameter semigroup is periodic if there exists $t_0>0$ such that $\Phi_{t_0} \equiv \mathrm{id}_X$.

Remark 5.2.2. The definition of one-parameter semigroup as a continuous map $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$ has as an immediate consequence the fact that also the map, still denoted by $\Phi$, from $\mathbb{R}^{+} \times X$ to $X$ sending $(t, z)$ in $\Phi_t(z)$ is continuous.

Remark 5.2.3. If $\Phi_{t_0} \equiv \mathrm{id}X$, then $\Phi{k t_0} \equiv \mathrm{id}X$ for all $k \in \mathbb{N}$. Furthermore, if $t>t_0$, writing $t=s+k t_0$ with $k=\left\lfloor t / t_0\right\rfloor \in \mathbb{N}$ and $s \in\left[0, t_0\right)$ we see that $\Phi_t \equiv \Phi_s$, and hence $\Phi$ is completely determined by $\Phi{\left[0, t_0\right]}$.

Our first result shows that not every function can be imbedded in a one-parameter semigroup

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups on Riemann surfaces

The aim of this section is to thoroughly investigate one-parameter semigroups on Riemann surfaces different from the unit disk, postponing the study of one-parameter semigroups on $\mathbb{D}$ to the remaining sections of this chapter.
Proposition 5.3.1. Let $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$ be a one-parameter semigroup on a Riemann surface $X$ with non-Abelian fundamental group. Then $\Phi$ is trivial.
Proof. By Theorem 2.6.2, we should have $\Phi_t \equiv \mathrm{id}_X$ for small $t$, and hence for all $t$.
So, we are left with just a few cases to investigate; let us start with the Riemann sphere.

Proposition 5.3.2. Let $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(\widehat{\mathbb{C}}, \widehat{\mathbb{C}})$ be a nontrivial one-parameter semigroup on the Riemann sphere $\widehat{\mathbb{C}}$. Then $\Phi$ extends to a one-parameter group, still denoted by $\Phi$, and there is $\gamma \in \operatorname{Aut}(\widehat{\mathbb{C}})$ such that either:
(i) $y^{-1} \circ \Phi_t \circ \gamma(z)=z+$ at for some $a \in \mathbb{C}^$, or (ii) $\gamma^{-1} \circ \Phi_t \circ \gamma(z)=e^{-b t} z$ for some $b \in \mathbb{C}^$.
In case (i), $\Phi$ has a unique fixed point with spectral value 0 and it is never periodic. In case (ii), $\Phi$ has two distinct fixed points with spectral value respectively $\pm b$; moreover, $\Phi$ is periodic if and only if $b \in \mathbb{R}^* i$ and then it has period $2 \pi /|b|$.

Proof. By Propositions 5.2.4 and 5.2.5, $\Phi$ extends to a one-parameter group, because the compactness of $\widehat{\mathbb{C}}$ implies that any injective holomorphic self-map of $\widehat{\mathbb{C}}$ is also surjective, and hence an automorphism.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups

5.2.1.定义设$X$为黎曼曲面。在$X$上的全纯映射的单参数半群(简称为单参数半群)是一个从$\mathbb{R}^{+}$到$\operatorname{Hol}(X, X)$的具有复合的连续半群同态$\Phi$。$X$上全纯映射的单参数群是从$(\mathbb{R},+)$到$\operatorname{Hol}(X, X)$的连续群同态。当$t \in \mathbb{R}^{+}$和$z \in X$时，我们经常写$\Phi_t(z)$或$\Phi(t, z)$而不是$\Phi(t)(z)$。平凡单参数半群是所有$t \in \mathbb{R}^{+}$的平凡同态$\Phi_t \equiv \operatorname{id}X$。最后，我们将说一个非平凡单参数半群是周期的，如果存在$t_0>0$使得$\Phi{t_0} \equiv \mathrm{id}_X$。

5.2.2.将单参数半群定义为连续映射$\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$的直接结果是，在$\Phi_t(z)$中发送$(t, z)$的从$\mathbb{R}^{+} \times X$到$X$的映射(仍然表示为$\Phi$)也是连续的。

5.2.3.如果是$\Phi_{t_0} \equiv \mathrm{id}X$，那么所有的$k \in \mathbb{N}$都是$\Phi{k t_0} \equiv \mathrm{id}X$。此外，如果$t>t_0$，用$k=\left\lfloor t / t_0\right\rfloor \in \mathbb{N}$和$s \in\left[0, t_0\right)$写$t=s+k t_0$，我们看到$\Phi_t \equiv \Phi_s$，因此$\Phi$完全由$\Phi{\left[0, t_0\right]}$决定。

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups on Riemann surfaces

(i) $y^{-1} \circ \Phi_t \circ \gamma(z)=z+$ at对于一些$a \in \mathbb{C}^$，或(ii) $\gamma^{-1} \circ \Phi_t \circ \gamma(z)=e^{-b t} z$对于一些$b \in \mathbb{C}^$。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

In the last section, we saw that parabolic self-maps of $\mathbb{D}$ fall in two categories having different dynamical behavior: positive hyperbolic step and zero hyperbolic step. So it is interesting to have some procedure to decide to which category a given parabolic self-map belongs.

In this section, we collect a few results of this kind, assuming a bit of regularity at the Wolff point. The main technical step is the following.

Proposition 4.7.1. Let $F \in \operatorname{Hol}\left(\mathrm{H}^{+}, \mathbb{H}^{+}\right)$be of the form $F(w)=w+i \alpha+\eta(w)$ with $\alpha \in \mathbb{C}$ and
$$\lim _{w \rightarrow \infty} \eta(w)=0$$
Then:
(i) F is parabolic with Wolff point at infinity;
(ii) $\frac{1}{v} F^v\left(w_0\right) \rightarrow$ i $\alpha$ as $v \rightarrow+\infty$ for every $w_0 \in \mathbb{H}^{+}$;
(iii) $\operatorname{Re} \alpha \geq 0$;
(iv) for each $w_0 \in \mathbb{H}^{+}$, the sequence $\left{\operatorname{Im} F^v\right.$ ( $\left.\left.w_0\right)\right}$ is not decreasing;
(v) if $\alpha=0$, then $F$ has zero hyperbolic step;
(vi) $F$ has zero hyperbolic step if and only if $\operatorname{Im} F^v\left(w_0\right) \rightarrow+\infty$ for some (and hence all) $w_0 \in \mathbb{H}^{+}$;
(vii) if $\operatorname{Re} \alpha>0$, then $F$ has zero hyperbolic step;
(viii) if $\alpha \neq 0$, then the orbit $\left{F^v\left(w_0\right)\right}$ tends to $\infty$ nontangentially if and only if $\operatorname{Re} \alpha>0$.

## 数学代写|黎曼曲面代写Riemann surface代考|Boundary fixed points

Recall that a boundary fixed point of a $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is a point $\sigma \in \partial \mathbb{D}$ such that $f(\sigma)=\sigma$, where $f(\sigma)$ is the nontangential limit of $f$ at $\sigma$ (see Definition 2.3.14). In Remark 2.3.15, we saw that if $\sigma$ is a boundary fixed point then we can define the derivative $f^{\prime}(\sigma)$ of $f$ at $\sigma$ by setting $f^{\prime}(\sigma)=\beta_f(\sigma) \in(0,+\infty]$; in particular, $f^{\prime}(\sigma)$ is the nontangential limit of $f^{\prime}$ at $\sigma$ when $\beta_f(\sigma)<+\infty$.

Definition 4.8.1. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be a holomorphic self-map of the unit disk. We say that $\sigma \in \partial \mathbb{D}$ is a boundary repelling fixed point if it is a boundary fixed point with $f^{\prime}(\sigma)>1$. Given $A>1$, we shall set
$$\operatorname{Fix}_A(f)=\left{\sigma \in \partial \mathbb{D} \mid f(\sigma)=\sigma \text { and } f^{\prime}(\sigma) \leq A\right}$$
Corollaries 2.3.16 and 2.5.5 say that if $f$ has a fixed point in $\mathbb{D}$, then all boundary fixed points are repelling, and that if $f$ has no fixed points in $\mathbb{D}$ then exactly one boundary fixed point is not repelling, the Wolff point of $f$. Furthermore, we have $f^{\prime}\left(\sigma_1\right) f^{\prime}\left(\sigma_2\right) \geq 1$ for all pairs of boundary fixed points (Theorem 2.3.13 contains a more precise estimate for boundary contact points).

In this section, we shall prove a precise quantitative generalization of these facts that we shall use in the next section to study the backward dynamics of a holomorphic self-map of $\mathbb{D}$.

We shall need two lemmas. The first one concerns Blaschke products (see Definition 1.5.5).

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

$$\lim _{w \rightarrow \infty} \eta(w)=0$$

(i) F在无穷远处具有Wolff点的抛物线;
(ii)对于每一个$w_0 \in \mathbb{H}^{+}$, $\frac{1}{v} F^v\left(w_0\right) \rightarrow$ I $\alpha$为$v \rightarrow+\infty$;
(iii) $\operatorname{Re} \alpha \geq 0$;
(iv)对于每个$w_0 \in \mathbb{H}^{+}$，顺序$\left{\operatorname{Im} F^v\right.$ ($\left.\left.w_0\right)\right}$)不递减;
(v)如果$\alpha=0$，则$F$的双曲步长为零;
(vi) $F$有零双曲阶跃当且仅当$\operatorname{Im} F^v\left(w_0\right) \rightarrow+\infty$对于一些(因此全部)$w_0 \in \mathbb{H}^{+}$;
(vii)若$\operatorname{Re} \alpha>0$，则$F$的双曲步长为零;
(viii)如果$\alpha \neq 0$，则轨道$\left{F^v\left(w_0\right)\right}$非切向$\infty$当且仅当$\operatorname{Re} \alpha>0$。

## 数学代写|黎曼曲面代写Riemann surface代考|Boundary fixed points

4.8.1.定义设$f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$为单位盘的全纯自映射。如果$\sigma \in \partial \mathbb{D}$与$f^{\prime}(\sigma)>1$为边界不动点，则称其为边界排斥不动点。给定$A>1$，我们将设置
$$\operatorname{Fix}_A(f)=\left{\sigma \in \partial \mathbb{D} \mid f(\sigma)=\sigma \text { and } f^{\prime}(\sigma) \leq A\right}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic dynamics

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic dynamics

Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D}) \backslash\left{\mathrm{id}{\mathbb{D}}\right}$ with Wolff point $\tau_f \in \overline{\mathbb{D}}$. If $\tau_f \in \mathbb{D}$, the Schwarz-Pick lemma implies that $\left|f^{\prime}\left(\tau_f\right)\right| \leq 1$, with equality if and only if $f$ is an elliptic automorphism. On the other hand, if $\tau_f \in \partial \mathbb{D}$ then Corollary 2.5.5 implies that $0{\mathbb{D}}\right}$ with Wolff point $\tau_f \in \overline{\mathbb{D}}$. We say that $f$ is:

• elliptic if $\tau_f \in \mathbb{D}$
• hyperbolic if $\tau_f \in \partial \mathbb{D}$ and $0<f^{\prime}\left(\tau_f\right)<1$;
• parabolic if $\tau_f \in \partial \mathbb{D}$ and $f^{\prime}\left(\tau_f\right)=1$.
Moreover, if $f$ is elliptic we shall say that it is attracting if $0<\left|f^{\prime}\left(\tau_f\right)\right|<1$ and that it is superattracting if $f^{\prime}\left(\tau_f\right)=0$.

We begin studying attracting elliptic functions, which is the easiest case. We shall see that the dynamics is modeled on the dynamics of the linear map $F(z)=f^{\prime}\left(\tau_f\right) z$; in particular, we shall obtain a model (in the sense of Definition 3.5.2) of the form $(\mathbb{C}, \psi, F)$ and we shall show that the orbits approach the Wolff point in a way comparable to the way the orbits of $F$ approach the origin. This is the content of the Kœnigs theorem.

## 数学代写|黎曼曲面代写Riemann surface代考|Superattracting dynamics

The superattracting elliptic case has slightly different features, mainly because the function $f$ is never injective in a neighborhood of its Wolff point, and thus it cannot have a model in the sense of Theorem 3.5.10. However, we shall still be able to change variables so that in the new coordinates $f$ will be expressed in a simple form; but in general it will not be possible to extend the coordinate map to the whole of $\mathbb{D}$. To express our results, we need a couple of definitions.
Definition 4.2.1. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ and let
$$f(z)=a_0+a_1\left(z-z_0\right)+a_2\left(z-z_0\right)^2+\cdots$$
be the power series expansion of $f$ at a point $z_0 \in \mathbb{D}$. The multiplicity $m_f^1\left(z_0\right)$ of $f$ at $z_0$ is given by $m_f^1\left(z_0\right)=\min \left{k \mid a_k \neq 0\right}$. More generally, given $v \geq 1$ the $v$-multiplicity $m_f^v\left(z_0\right)$ of $f$ at $z_0$ is the multiplicity of $f^v$ at $z_0$, i. e., $m_f^v\left(z_0\right)=m_{f^v}^1\left(z_0\right)$.

Clearly, we have $f(0)=0$ if and only if $m_f^1(0) \geq 1$ and 0 is superattracting if and only if $m_f^1(0) \geq 2$.

We shall now prove the superattracting version of Theorem 4.1.2, the Böttcher theorem.

Theorem 4.2.2 (Böttcher, 1904). Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be superattracting elliptic. Let $\tau_f \in \mathbb{D}$ be its Wolff point, and $m \geq 2$ the multiplicity of $f-\tau_f$ at $\tau_f$. Then:
(i) there exists a simply connected $f$-absorbing domain $A \subset \mathbb{D}$ containing $\tau_f$ and $a$ never vanishing holomorphic function $\psi \in \operatorname{Hol}(A, \mathbb{C})$ with $\psi\left(\tau_f\right)=1$ such that the function $\varphi(z)=z \psi(z)$ is the unique solution of the functional equation
$$\varphi \circ f(z)=\varphi(z)^m$$
satisfying $\varphi\left(\tau_f\right)=0$ and $\varphi^{\prime}\left(\tau_f\right)=1$;
(ii) for every $z \in A \backslash\left{\tau_f\right}$, we have
$$\lim _{v \rightarrow+\infty}\left[\frac{f^{v+1}(z)-\tau_f}{f^v(z)-\tau_f}\right]^{1 / m^v}=\varphi(z)^{m-1} .$$
Proof. As we have seen in the proof of Theorem 4.1.2, recalling in particular (4.4), without loss of generality we can assume that $\tau_f=0$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Superattracting dynamics

4.2.1.定义让$f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$和让
$$f(z)=a_0+a_1\left(z-z_0\right)+a_2\left(z-z_0\right)^2+\cdots$$

(1)存在一个含有$\tau_f$和$a$不灭全纯函数$\psi \in \operatorname{Hol}(A, \mathbb{C})$与$\psi\left(\tau_f\right)=1$的单连通$f$吸收域$A \subset \mathbb{D}$，使得函数$\varphi(z)=z \psi(z)$是泛函方程的唯一解
$$\varphi \circ f(z)=\varphi(z)^m$$

(ii)对于每一个$z \in A \backslash\left{\tau_f\right}$，我们有
$$\lim _{v \rightarrow+\infty}\left[\frac{f^{v+1}(z)-\tau_f}{f^v(z)-\tau_f}\right]^{1 / m^v}=\varphi(z)^{m-1} .$$

## 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。

## 数学代写|黎曼曲面代写Riemann surface代考|The Burns-Krantz theorem

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|The Burns-Krantz theorem

In Example 2.5.7, we noticed that we can find a function $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D}) \backslash \operatorname{Aut}(\mathbb{D})$ such that $f(1)=1$ and $f^{\prime}(1)=1$. This in sharp contrast with the uniqueness part of the Schwarz lemma, which says that if $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is such that $f(0)=0$ and $f^{\prime}(0)=1$ then $f \equiv \mathrm{id}_{\mathbb{D}}$. This section is devoted to finding a satisfying boundary version of the uniqueness part of the Schwarz lemma.

A possible reformulation of the uniqueness part of the Schwarz lemma is the following: if $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is such that $f(z)=z+o(|z|)$, then $f \equiv \mathrm{id}{\mathbb{D}}$. To get a boundary version of this statement, one might look for $c>0$ such that if $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is such that $f(z)=z+o\left(|\sigma-z|^c\right)$ as $z \rightarrow \sigma \in \partial \mathbb{D}$ then $f \equiv \mathrm{id}{\mathbb{D}}$. Example 2.5.7 (see also Example 2.7.5 below) shows that $c$ must be necessarily at least 3 ; indeed, we shall prove (Theorem 2.7.4) that the best value of $c$ is exactly 3. To do so, we shall start from the (more invariant) uniqueness part of the Schwarz-Pick lemma Corollary 1.1.16 that can be stated as follows: if $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is such that
$$\left|f^h(z)\right|=\left|f^{\prime}(z)\right| \frac{1-|z|^2}{1-|f(z)|^2}=1+o(1)$$
as $z \rightarrow z_0 \in \mathbb{D}$ then $f \in \operatorname{Aut}(\mathbb{D})$. In Theorem 2.7.2, we shall show that if $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is such that
$$\left|f^h(z)\right|=1+o\left(|\sigma-z|^2\right)$$
as $z \rightarrow \sigma \in \partial \mathbb{D}$ then $f \in \operatorname{Aut}(\mathbb{D})$, and the exponent 2 is optimal. From this, it will not be too difficult to deduce Theorem 2.7.4.

## 数学代写|黎曼曲面代写Riemann surface代考|Discrete dynamics on Riemann surfaces

In this chapter, we begin to deal with the main argument of this book: holomorphic dynamics. As anticipated in the Introduction (and in the title of the book), we shall mainly deal with hyperbolic Riemann surfaces, where the whole strength of the Montel theorem is available. The idea is that if $X$ is a hyperbolic Riemann surface and $f \in \operatorname{Hol}(X, X)$ then the sequence of iterates of $f$ is a normal family and so its behavior cannot be chaotic. For this reason, holomorphic dynamics on hyperbolic Riemann surfaces is completely different from holomorphic dynamics on elliptic Riemann surfaces (i. e., $\widehat{\mathbb{C}}$ ) or parabolic Riemann surfaces (e. g., $\mathbb{C}$ ), where a large part of the theory is devoted to studying the chaotic part of the dynamics, concentrated on the so-called Julia set. On hyperbolic Riemann surfaces, the Julia set is empty: indeed, we shall be able to prove that (with a few exceptions completely classified in the case of automorphisms) the sequence of iterates of a holomorphic self-map of a hyperbolic Riemann surface either is compactly divergent or converges, uniformly on compact sets, to a constant.

This is the best result of this kind for a generic hyperbolic Riemann surface. But if $D \subset \widehat{X}$ is a hyperbolic domain then we can say something more. In this case, in fact, $\operatorname{Hol}(D, D)$ is contained in $\operatorname{Hol}(D, \widehat{X})$, a space without compactly divergent sequences; therefore, the sequence of iterates of a function $f \in \operatorname{Hol}(D, D)$ is relatively compact in $\operatorname{Hol}(D, \widehat{X})$ and so it always has converging subsequences, converging possibly to a point of $\partial D$

This observation (already somewhat anticipated in Proposition 1.7.20) leads to the core of this chapter: the Heins theorem, stating that if $D \subset \widehat{X}$ is a hyperbolic domain of regular type and $f \in \operatorname{Hol}(D, D)$ is not an automorphism then the sequence of iterates of $f$ converges, uniformly on compact sets, to a constant $\tau \in \bar{D}$, the Wolff point of $f$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|The Burns-Krantz theorem

$$\left|f^h(z)\right|=\left|f^{\prime}(z)\right| \frac{1-|z|^2}{1-|f(z)|^2}=1+o(1)$$

$$\left|f^h(z)\right|=1+o\left(|\sigma-z|^2\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。