## 数学代写|复分析作业代写Complex function代考|MATH3979

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Biholomorphic Mappings of the Complex Plane to Itself

The simplest open subset of $\mathbb{C}$ is $\mathbb{C}$ itself. Thus it is natural to begin our study of conformal mappings by considering the biholomorphic mappings of $\mathbb{C}$ to itself. Of course, there are a great many holomorphic functions from $\mathbb{C}$ to $\mathbb{C}$, but rather few of these turn out to be one-to-one and onto. The techniques that we use to analyze even this rather simple situation will introduce some of the basic ideas in the study of mappings. The biholomorphic mappings from $\mathbb{C}$ to $\mathbb{C}$ can be explicitly described as follows:

Theorem 6.1.1. A function $f: \mathbb{C} \rightarrow \mathbb{C}$ is a conformal mapping if and only if there are complex numbers $a, b$ with $a \neq 0$ such that
$$f(z)=a z+b, \quad z \in \mathbb{C} .$$
One aspect of the theorem is fairly obvious: If $a, b \in \mathbb{C}$ and $a \neq 0$, then the map $z \mapsto a z+b$ is certainly a conformal mapping of $\mathbb{C}$ to $\mathbb{C}$. In fact one checks easily that $z \mapsto(z-b) / a$ is the inverse mapping. The interesting part of the theorem is that these are in fact the only conformal maps of $\mathbb{C}$ to $\mathbb{C}$. To see this, fix a conformal map of $f: \mathbb{C} \rightarrow \mathbb{C}$. We begin with some lemmas:
Lemma 6.1.2. The holomorphic function $f$ satisfies
$$\lim _{|z| \rightarrow+\infty}|f(z)|=+\infty \text {. }$$
That is, given $\epsilon>0$, there is a number $C>0$ such that if $|z|>C$, then $|f(z)|>1 / \epsilon$

Proof. This is a purely topological fact, and our proof uses no complex analysis as such.

The set ${z:|z| \leq 1 / \epsilon}$ is a compact subset of $\mathbb{C}$. Since $f^{-1}: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic, it is continuous. Also the continuous image of a compact set is compact. Therefore $S=f^{-1}({z:|z| \leq 1 / \epsilon})$ is compact. By the Heine-Borel theorem (see [RUD1]), $S$ must be bounded. Thus there is a positive number $C$ such that $S \subseteq{z:|z| \leq C}$.

## 数学代写|复分析作业代写Complex function代考|Biholomorphic Mappings of the Unit Disc to Itself

In this section the set of all conformal maps of the unit disc to itself will be determined. The determination process is somewhat less natural than in the last section, for the reader is presented with a “list” of mappings, and then it is proved that these are all the conformal self-maps of the disc (i.e., conformal maps of the disc to itself). This artificiality is a bit unsatisfying; later, when we treat the geometric structure known as the Bergman metric (Chapter 14), we shall be able to explain the genesis of these mappings.
Our first step is to determine those conformal maps of the disc to the disc that fix the origin. Let $D$ denote the unit disc.

Lemma 6.2.1. A holomorphic function $f: D \rightarrow D$ that satisfies $f(0)=0$ is a conformal mapping of $D$ onto itself if and only if there is a complex number $\omega$ with $|\omega|=1$ such that
$$f(z) \equiv \omega z \text { for all } z \in D .$$
In other words, a conformal self-map of the disc that fixes the origin must be a rotation.

Proof. If $\omega \in \mathbb{C}$ and $|\omega|=1$, then clearly the function $f(z) \equiv \omega z$ is a conformal self-map of the disc: The inverse mapping is $z \mapsto z / \omega$.

To prove the converse, suppose that $f: D \rightarrow D$ is a conformal self-map of the disc that fixes the origin. Let $g=f^{-1}: D \rightarrow D$. By the Schwarz lemma (Proposition 5.5.1),
$$\left|f^{\prime}(0)\right| \leq 1 \text { and }\left|g^{\prime}(0)\right| \leq 1 \text {. }$$

## 数学代写|复分析作业代写Complex function代考|Biholomorphic Mappings of the Complex Plane to Itself

$$f(z)=a z+b, \quad z \in \mathbb{C} .$$

$$\lim _{|z| \rightarrow+\infty}|f(z)|=+\infty$$

## 数学代写|复分析作业代写Complex function代考|Biholomorphic Mappings of the Unit Disc to Itself

$$f(z) \equiv \omega z \text { for all } z \in D .$$

$$\left|f^{\prime}(0)\right| \leq 1 \text { and }\left|g^{\prime}(0)\right| \leq 1$$

## 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|复分析作业代写Complex function代考|MTH3019

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|The Schwarz Lemma

This section treats certain estimates that bounded holomorphic functions on the unit disc necessarily satisfy. At first sight, these estimates appear to be restricted to such a specific situation that they are of limited interest. But even this special situation occurs so often that the estimates are in fact very useful. Moreover, it was pointed out by Ahlfors [AHL1] that these estimates can be interpreted as a statement about certain kinds of geometric structures that occur in many different contexts in complex analysis. A treatment of this point of view that is accessible to readers who have reached this point in the present book can be found in [KRA3]. This section presents the classical, analytic viewpoint in the subject.

Proposition 5.5.1 (Schwarz’s lemma). Let $f$ be holomorphic on the unit disc. Assume that
(1) $|f(z)| \leq 1$ for all $z$,
(2) $f(0)=0$.
Then $|f(z)| \leq|z|$ and $\left|f^{\prime}(0)\right| \leq 1$.
If either $|f(z)|=|z|$ for some $z \neq 0$ or if $\left|f^{\prime}(0)\right|=1$, then $f$ is a rotation: $f(z) \equiv \alpha z$ for some complex constant $\alpha$ of unit modulus.

Proof. Consider the function $g(z)=f(z) / z$. This function is holomorphic on $D(0,1) \backslash{0}$. Also $\lim _{z \rightarrow 0} g(z)=f^{\prime}(0)$. So if we define $g(0)=f^{\prime}(0)$, then $g$ is continuous on $D(0,1)$. By the Riemann removable singularities theorem (Theorem 4.1.1), $g$ is then holomorphic on all of $D(0,1)$.

Restrict attention to the closed $\operatorname{disc} \bar{D}(0,1-\epsilon)$ for $\epsilon>0$ and small. On the boundary of this disc, $|g(z)| \leq 1 /(1-\epsilon)$. The maximum modulus theorem then implies that $|g(z)| \leq 1 /(1-\epsilon)$ on this entire disc $D(0,1-\epsilon)$. Letting $\epsilon \rightarrow 0^{+}$then yields that $|g(z)| \leq 1$ on $D=D(0,1)$. In other words, $|f(z)| \leq|z|$. Now $g(0)=f^{\prime}(0)$ so we also see that $\left|f^{\prime}(0)\right| \leq 1$. That completes the proof of the first half of the theorem.

If $|f(z)|=|z|$ for some $z \neq 0$, then $|g(z)|=1$. Since $|g(z)| \leq 1$ on the entire disc, we conclude from the maximum modulus principle that $g(z)$ is a constant of modulus 1 . Let $\alpha$ be that constant. Then $f(z) \equiv \alpha z$.

## 数学代写|复分析作业代写Complex function代考|Holomorphic Functions as Geometric Mappings

Like Chapter 5 , this chapter is concerned primarily with geometric questions. While the proofs that we present will of course be analytic, it is useful to interpret them pictorially. The proofs of the theorems presented here arose from essentially pictorial ideas, and these pictures can still serve to guide our perceptions. In fact geometry is a pervasive part of the subject of complex analysis and will occur in various forms throughout the remainder of the book.

The main objects of study in this chapter are holomorphic functions $h: U \rightarrow V$, with $U, V$ open in $\mathbb{C}$, that are one-to-one and onto. Such a holomorphic function is called a conformal (or biholomorphic) mapping. The fact that $h$ is supposed to be one-to-one implies that $h^{\prime}$ is nowhere zero on $U$ [remember, by Theorem $5.2 .2$, that if $h^{\prime}$ vanishes to order $k \geq 0$ at a point $P \in U$, then $h$ is $(k+1)$-to-1 in a small neighborhood of $P$ ]. As a result, $h^{-1}: V \rightarrow U$ is also holomorphic (see Section 5.2). A conformal map $h: U \rightarrow V$ from one open set to another can be used to transfer holomorphic functions on $U$ to $V$ and vice versa: That is, $f: V \rightarrow \mathbb{C}$ is holomorphic if and only if $f \circ h$ is holomorphic on $U$; and $g: U \rightarrow \mathbb{C}$ is holomorphic if and only if $g \circ h^{-1}$ is holomorphic on $V$.

Thus, if there is a conformal mapping from $U$ to $V$, then $U$ and $V$ are essentially indistinguishable from the viewpoint of complex function theory. On a practical level, one can often study holomorphic functions on a rather complicated open set by first mapping that open set to some simpler open set, then transferring the holomorphic functions as indicated.

## 数学代写|复分析作业代写Complex function代考|The Schwarz Lemma

(1) $|f(z)| \leq 1$ 对所有人 $z$ ，
(2) $f(0)=0$.

## 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|复分析作业代写Complex function代考|MATH307

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Further Results on the Zeros of Holomorphic Functions

In the previous sections of this chapter, we have developed a detailed understanding of the local behavior of holomorphic functions, that is, of their behavior in a small neighborhood of a particular point. The methods we used, and especially Proposition 5.1.2, can be applied in a wider context to the “global behavior” of a holomorphic function on its whole domain of definition. In this section we state and prove two important results of this sort.

Theorem 5.3.1 (Rouché’s theorem). Suppose that $f, g: U \rightarrow \mathbb{C}$ are holomorphic functions on an open set $U \subseteq \mathbb{C}$. Suppose also that $\bar{D}(P, r) \subseteq U$ and that, for each $\zeta \in \partial D(P, r)$,
$$|f(\zeta)-g(\zeta)|<|f(\zeta)|+|g(\zeta)|$$
Then
$$\frac{1}{2 \pi i} \oint_{\partial D(P, r)} \frac{f^{\prime}(\zeta)}{f(\zeta)} d \zeta=\frac{1}{2 \pi i} \oint_{\partial D(P, r)} \frac{g^{\prime}(\zeta)}{g(\zeta)} d \zeta$$

That is, the number of zeros of $f$ in $D(P, r)$ counting multiplicities equals the number of zeros of $g$ in $D(P, r)$ counting multiplicities.

Before beginning the proof of Rouché’s theorem, we note that the (at first strange looking) inequality (*) implies that neither $f(\zeta)$ nor $g(\zeta)$ can vanish on $\partial D(P, r)$. In particular, neither $f$ nor $g$ vanishes identically; moreover, the integrals of $f^{\prime} / f$ and of $g^{\prime} / g$ on $\partial D(P, r)$ are defined.

Also, $()$ implies that the function $f(\zeta) / g(\zeta)$ cannot take a value in ${x+i 0: x \leq 0}$ for any $\zeta \in \partial D(P, r)$. If it did, say $$\frac{f(\zeta)}{g(\zeta)}=\lambda \leq 0$$ for some $\zeta \in \partial D(P, r)$, then \begin{aligned} \left|\frac{f(\zeta)}{g(\zeta)}-1\right| &=|\lambda-1| \ &=-\lambda+1 \ &=\left|\frac{f(\zeta)}{g(\zeta)}\right|+1 \end{aligned} hence $$|f(\zeta)-g(\zeta)|=|f(\zeta)|+|g(\zeta)|$$ This equality contradicts $()$.

## 数学代写|复分析作业代写Complex function代考|The Maximum Modulus Principle

Consider the $C^{\infty}$ function $g$ on the unit disc given by $g(z)=2-|z|^{2}$. Notice that $1<|g(z)| \leq 2$ and that $g(0)=2$. The function assumes an interior maximum at $z=0$. One of the most startling features of holomorphic functions is that they cannot behave in this fashion: In stating the results about this phenomenon, the concept of a connected open set occurs so often that it is convenient to introduce a single word for it.

Definition 5.4.1. A domain in $\mathbb{C}$ is a connected open set. A bounded domain is a connected open set $U$ such that there is an $R>0$ with $|z|<R$ for all $z \in U$.

Theorem 5.4.2 (The maximum modulus principle). Let $U \subseteq \mathbb{C}$ be a domain. Let $f$ be a holomorphic function on $U$. If there is a point $P \in U$ such that $|f(P)| \geq|f(z)|$ for all $z \in U$, then $f$ is constant.

Proof. Assume that there is such a $P$. If $f$ is not constant, then $f(U)$ is open by the open mapping principle. Hence there are points $\zeta$ of $f(U)$ with $|\zeta|>|f(P)|$. This is a contradiction. Hence $f$ is a constant.

Here is a consequence of the maximum modulus principle that is often useful:

Corollary 5.4.3 (Maximum modulus theorem). Let $U \subseteq \mathbb{C}$ be a bounded domain. Let $f$ be a continuous function on $\bar{U}$ that is holomorphic on $U$. Then the maximum value of $|f|$ on $\bar{U}$ (which must occur, since $\bar{U}$ is closed and bounded) must occur on $\partial U$.

## 数学代写|复分析作业代写Complex function代考|Further Results on the Zeros of Holomorphic Functions

$$|f(\zeta)-g(\zeta)|<|f(\zeta)|+|g(\zeta)|$$

$$\frac{1}{2 \pi i} \oint_{\partial D(P, r)} \frac{f^{\prime}(\zeta)}{f(\zeta)} d \zeta=\frac{1}{2 \pi i} \oint_{\partial D(P, r)} \frac{g^{\prime}(\zeta)}{g(\zeta)} d \zeta$$

$$\frac{f(\zeta)}{g(\zeta)}=\lambda \leq 0$$

$$\left|\frac{f(\zeta)}{g(\zeta)}-1\right|=|\lambda-1| \quad=-\lambda+1=\left|\frac{f(\zeta)}{g(\zeta)}\right|+1$$

$$|f(\zeta)-g(\zeta)|=|f(\zeta)|+|g(\zeta)|$$

## 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|复分析作业代写Complex function代考|KMA152

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Counting Zeros and Poles

Suppose that $f: U \rightarrow \mathbb{C}$ is a holomorphic function on a connected, open set $U \subseteq \mathbb{C}$ and that $\bar{D}(P, r) \subseteq U$. We know from the Cauchy integral formula that the values of $f$ on $D(P, r)$ are completely determined by the values of $f$ on $\partial D(P, r)$. In particular, the number and even the location of the zeros of $f$ in $D(P, r)$ are determined in principle by $f$ on $\partial D(P, r)$. But it is nonetheless a pleasant surprise that there is a simple formula for the number of zeros of $f$ in $D(P, r)$ in terms of $f$ (and $f^{\prime}$ ) on $\partial D(P, r)$. In order to construct this formula, we shall have to agree to count zeros in a particular fashion. This method of counting will in fact be a generalization of the notion of counting the zeros of a polynomial according to multiplicity. We now explain the precise idea.

Let $f: U \rightarrow \mathbb{C}$ be holomorphic as before, and assume that $f$ has zeros but that $f$ is not identically zero. Fix $z_{0} \in U$ such that $f\left(z_{0}\right)=0$. Since the zeros of $f$ are isolated, there is an $r>0$ such that $\bar{D}\left(z_{0}, r\right) \subseteq U$ and such that $f$ does not vanish on $\bar{D}\left(z_{0}, r\right) \backslash\left{z_{0}\right}$.

Now the power series expansion of $f$ about $z_{0}$ has a first nonzero term determined by the least positive integer $n$ such that $f^{(n)}\left(z_{0}\right) \neq 0$. [Note that $n \geq 1$ since $f\left(z_{0}\right)=0$ by hypothesis.] Thus the power series expansion of $f$ about $z_{0}$ begins with the $n^{\text {th }}$ term:
$$f(z)=\sum_{j=n}^{\infty} \frac{1}{j !} \frac{\partial^{j} f}{\partial z^{j}}\left(z_{0}\right)\left(z-z_{0}\right)^{j} .$$
Under these circumstances we say that $f$ has a zero of order $n$ (or multiplicity $n$ ) at $z_{0}$. When $n=1$, then we say that $z_{0}$ is a simple zero of $f$.

## 数学代写|复分析作业代写Complex function代考|The Local Geometry of Holomorphic Functions

The argument principle for holomorphic functions (the formula of Proposition 5.1.2) has a consequence which is one of the most important facts about holomorphic functions considered as geometric mappings:

Theorem 5.2.1 (The open mapping theorem). If $f: U \rightarrow \mathbb{C}$ is a nonconstant holomorphic function on a connected open set $U$, then $f(U)$ is an open set in $\mathbb{C}$.

Before beginning the proof of the theorem, we discuss its significance. The theorem says, in particular, that if $U \subseteq \mathbb{C}$ is connected and open and if $f: U \rightarrow \mathbb{C}$ is holomorphic, then either $f(U)$ is a connected open set (the nonconstant case) or $f(U)$ is a single point. There is no analogous result for $C^{\infty}$, or even real analytic functions from $\mathbb{C}$ to $\mathbb{C}$ (or from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ ). As an example, consider the function
\begin{aligned} g: \mathbb{C} & \rightarrow \mathbb{C} \ z & \mapsto|z|^{2} . \end{aligned}
The domain of $g$ is the entire plane $\mathbb{C}$, which is certainly open and connected. The set $g(\mathbb{C})$, however, is ${x+i 0: \mathbb{R} \ni x \geq 0}$ which is not open as a subset of $\mathbb{C}$. The function $g$ is in fact real analytic, but of course not holomorphic.
Note, by contrast, that the holomorphic function
\begin{aligned} g: \mathbb{C} & \rightarrow \mathbb{C} \ z & \mapsto z^{2} \end{aligned}
has image the entire complex plane (which is, of course, an open set). More significantly, every open subset of $\mathbb{C}$ has image under $g$ which is open.
In the subject of topology, a function $f$ is defined to be continuous if the inverse image of any open set under $f$ is also open. In contexts where the $\epsilon-\delta$ definition makes sense, the $\epsilon-\delta$ definition is equivalent to the inverse-image-of-open-sets definition.

## 数学代写|复分析作业代写Complex function代考|Counting Zeros and Poles

$$f(z)=\sum_{j=n}^{\infty} \frac{1}{j !} \frac{\partial^{j} f}{\partial z^{j}}\left(z_{0}\right)\left(z-z_{0}\right)^{j} .$$

## 数学代写|复分析作业代写Complex function代考|The Local Geometry of Holomorphic Functions

$$g: \mathbb{C} \rightarrow \mathbb{C} z \quad \mapsto|z|^{2} .$$

$$g: \mathbb{C} \rightarrow \mathbb{C} z \quad \mapsto z^{2}$$

## 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|复分析作业代写Complex function代考|MATH3401

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Real and Holomorphic Antiderivatives

In this section we want to treat in greater generality the question of whether a real-valued harmonic function $u$ is the real part of a holomorphic function $F$. Notice that if we write $F=u+i v$, then the Cauchy-Riemann equations say that
$$\begin{gathered} \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} \ \frac{\partial v}{\partial y}=\frac{\partial u}{\partial x} \end{gathered}$$
In short, once $u$ is given, then $\partial v / \partial x$ and $\partial v / \partial y$ are completely determined. These in turn determine $v$ up to an additive constant. Thus determining the existence of $v$ (and hence of $F$ ) amounts to solving a familiar problem of multivariable calculus: Given two functions $f$ and $g$ (in this case $-\partial u / \partial y$ and $\partial u / \partial x$, respectively), can we find a function $v$ such that $\partial v / \partial x=f$ and $\partial v / \partial y=g ?$

A partial solution to this problem is given by the following theorem. We shall see later that the practice, begun in this theorem, of restricting consideration to functions defined on rectangles is not simply a convenience. In fact, the next theorem would actually be false if we considered functions defined on arbitrary open sets in $\mathbb{C}$ (see Exercise 52 ).

## 数学代写|复分析作业代写Complex function代考|Real and Complex Line Integrals

In the previous chapter, we approached the question of finding a function with given partial derivatives by integrating along vertical and horizontal directions only. The fact that the horizontal derivative is $\partial / \partial x$ and the vertical derivative is $\partial / \partial y$ then made the computations in Section $1.5$ obvious. But the restriction to such integrals is geometrically unnatural. In this section we are going to develop an integration process along more general curves. It is in fact not a new method of integration at all but is the process of line integration which you learned in calculus. Our chief job here is to make it rigorous and to introduce notation that is convenient for complex analysis.

First, let us define the class of curves we shall consider. It is convenient to think of a curve as a (continuous) function $\gamma$ from a closed interval $[a, b] \subseteq \mathbb{R}$ into $\mathbb{R}^{2} \approx \mathbb{C}$. Although it is frequently convenient to refer to the geometrical object $\tilde{\gamma} \equiv{\gamma(t): t \in[a, b]}$, most of our analysis will be done with the function $\gamma$. It is often useful to write
$$\gamma(t)=\left(\gamma_{1}(t), \gamma_{2}(t)\right) \quad \text { or } \quad \gamma(t)=\gamma_{1}(t)+i \gamma_{2}(t),$$
depending on the context. The curve $\gamma$ is called closed if $\gamma(a)=\gamma(b)$. It is called simple closed if $\left.\gamma\right|_{[a, b)}$ is one-to-one and $\gamma(a)=\gamma(b)$. Intuitively, a simple closed curve is a curve with no self-intersections, except of course for the closing up at $t=a, t=b$.

In order to work effectively with $\gamma$, we need to impose on it some differentiability properties. Since $\gamma$ is defined on a closed interval, this requires a new definition.

## 数学代写|复分析作业代写Complex function代考|Real and Holomorphic Antiderivatives

$$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} \frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}$$

## 数学代写|复分析作业代写Complex function代考|Real and Complex Line Integrals

$$\gamma(t)=\left(\gamma_{1}(t), \gamma_{2}(t)\right) \quad \text { or } \quad \gamma(t)=\gamma_{1}(t)+i \gamma_{2}(t),$$

## 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|复分析作业代写Complex function代考|MATH2521

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Complex Polynomials

In the calculus of real variables, polynomials are the simplest nontrivial functions. The purpose of this section is to consider complex-valued polynomials of a complex variable, with the idea of seeing what new features appear. Later we shall use the discussion as motivation for considering more general functions.

There are several slightly different ways of looking at polynomials from the complex viewpoint. One way is to consider polynomials in $x$ and $y$, $(x, y) \in \mathbb{R}^{2}$, with complex coefficients: for example, $(2+i) x y+3 i y^{2}+5 x^{2}$. Such polynomials give functions from $\mathbb{R}^{2}$ to $\mathbb{C}$, which we could equally well think of as functions from $\mathbb{C}$ to $\mathbb{C}$, with $(x, y)$ determined by $z=x+$ $i y$. Another kind of polynomial that we can consider is complex-coefficient polynomials in the complex variable $z$, for example, $i+(3+i) z+5 z^{2}$. These also give functions from $\mathbb{C}$ to $\mathbb{C}$. A polynomial in $z$ gives rise naturally to a polynomial in $x$ and $y$ by substituting $z=x+i y$ and expanding. For instance
\begin{aligned} i+(3+i) z+5 z^{2} &=i+(3+i)(x+i y)+5(x+i y)^{2} \ &=i+3 x-y+i x+3 i y+5 x^{2}+10 i x y-5 y^{2} \ &=i+(3+i) x+(3 i-1) y+5 x^{2}+(10 i) x y-5 y^{2} \end{aligned}
It is an important and somewhat surprising fact that the converse of this expansion process does not always work: there are many polynomials in $x$ and $y$ that cannot be written as polynomials in $z$. Let us consider a specific simple example: the polynomial $x$ itself. If it were true that
$$x=P(z)=P(x+i y)$$
for some polynomial $P(z)$ in $z$, then $P$ would have to be of first degree. But a first degree polynomial $a z+b=a x+i a y+b$ cannot be identically equal to $x$, no matter how we choose $a$ and $b$ in $\mathbb{C}$ (see Exercise 35 ). What is really going on here?

## 数学代写|复分析作业代写Complex function代考|Holomorphic Functions, the Cauchy-Riemann

Functions $f$ which satisfy $(\partial / \partial \bar{z}) f \equiv 0$ are the main concern of complex analysis. We make a precise definition:

Definition 1.4.1. A continuously differentiable $\left(C^{1}\right)$ function $f: U \rightarrow \mathbb{C}$ defined on an open subset $U$ of $\mathbb{C}$ is said to be holomorphic if
$$\frac{\partial f}{\partial \bar{z}}=0$$
at every point of $U$.
Remark: Some books use the word “analytic” instead of “holomorphic.” Still others say “differentiable” or “complex differentiable” instead of “holomorphic.” The use of “analytic” derives from the fact that a holomorphic function has a local power series expansion about each point of its domain. The use of “differentiable” derives from properties related to the CauchyRiemann equations and conformality. These pieces of terminology, and their significance, will all be sorted out as the book develops.

If $f$ is any complex-valued function, then we may write $f=u+i v$, where $u$ and $v$ are real-valued functions. For example,
$$z^{2}=\left(x^{2}-y^{2}\right)+i(2 x y)$$
in this example $u=x^{2}-y^{2}$ and $v=2 x y$. The following lemma reformulates Definition $1.4 .1$ in terms of the real and imaginary parts of $f$ :

Lemma 1.4.2. A continuously differentiable function $f: U \rightarrow \mathbb{C}$ defined on an open subset $U$ of $\mathbb{C}$ is holomorphic if, writing $f(z)=u(x, y)+i v(x, y)$, with $z=x+i y$ and real-valued functions $u$ and $v$, we have that $u$ and $v$ satisfy the equations
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$
at every point of $U$.

## 数学代写|复分析作业代写Complex function代考|Complex Polynomials

$i+(3+i) z+5 z^{2}$. 这些也给出了函数 $\mathbb{C}$ 至 C. 多项式在 $z$ 自然产生多项式 $x$ 和 $y$ 通过替换 $z=x+i y$ 和扩大。例 如
$$i+(3+i) z+5 z^{2}=i+(3+i)(x+i y)+5(x+i y)^{2} \quad=i+3 x-y+i x+3 i y+5 x^{2}+10 i x y$$

$$x=P(z)=P(x+i y)$$

## 数学代写|复分析作业代写Complex function代考|Holomorphic Functions, the Cauchy-Riemann

$$\frac{\partial f}{\partial \bar{z}}=0$$

$$z^{2}=\left(x^{2}-y^{2}\right)+i(2 x y)$$

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

## 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|复分析作业代写Complex function代考|MATH3979

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function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。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function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|复分析作业代写Complex function代考|Elementary Properties of the Complex Numbers

We take for granted the real numbers, which will be denoted by the symbol $\mathbb{R}$. Then we set $\mathbb{R}^{2}={(x, y): x \in \mathbb{R}, y \in \mathbb{R}}$. The complex numbers $\mathbb{C}$ consist of $\mathbb{R}^{2}$ equipped with some special algebraic operations. Namely, one defines
\begin{aligned} (x, y)+\left(x^{\prime}, y^{\prime}\right) &=\left(x+x^{\prime}, y+y^{\prime}\right), \ (x, y) \cdot\left(x^{\prime}, y^{\prime}\right) &=\left(x x^{\prime}-y y^{\prime}, x y^{\prime}+y x^{\prime}\right) . \end{aligned}
You can check for yourself that these operations of $+$ and – are commutative and associative.

It is both conventional and convenient to denote $(1,0)$ by 1 and $(0,1)$ by $i$. We also adopt the convention that, if $\alpha \in \mathbb{R}$, then
$$\alpha \cdot(x, y)=(\alpha, 0) \cdot(x, y)=(\alpha x, \alpha y) .$$
Then every complex number $(x, y)$ can be written in one and only one way in the form $x \cdot 1+y \cdot i$ with $x, y \in \mathbb{R}$. We usually write the number even more succinctly as $x+i y$. Then our laws of addition and multiplication become
\begin{aligned} (x+i y)+\left(x^{\prime}+i y^{\prime}\right) &=\left(x+x^{\prime}\right)+i\left(y+y^{\prime}\right), \ (x+i y) \cdot\left(x^{\prime}+i y^{\prime}\right) &=\left(x x^{\prime}-y y^{\prime}\right)+i\left(x y^{\prime}+y x^{\prime}\right) . \end{aligned}
Observe that $i \cdot i=-1$. Moreover, our multiplication law is consistent with the real multiplication introduced in line $(*)$.

The symbols $z, w, \zeta$ are frequently used to denote complex numbers. Unless it is explicitly stated otherwise, we always take $z=x+i y, w=$ $u+i v, \zeta=\xi+i \eta$. The real number $x$ is called the real part of $z$ and is written $x=\operatorname{Re} z$. Likewise $y$ is called the imaginary part of $z$ and is written $y=\operatorname{Im} z$.

## 数学代写|复分析作业代写Complex function代考|Further Properties of the Complex Numbers

We first consider the complex exponential, which we define as follows:
(1) If $z=x$ is real, then
$$e^{z}=e^{x} \equiv \sum_{j=0}^{\infty} \frac{x^{j}}{j !}$$
as in calculus.
(2) If $z=i y$ is pure imaginary, then
$$e^{z}=e^{i y} \equiv \cos y+i \sin y .$$
(3) If $z=x+i y$, then
$$e^{z}=e^{x+i y} \equiv e^{x} \cdot(\cos y+i \sin y) .$$
Parts (2) and (3) of the definition, due to Euler, may seem somewhat arbitrary. We shall now show, using power series, that these definitions are perfectly natural. We shall wait until Section $3.2$ to give a careful presentation of the theory of complex power series. So the power series arguments that we are about to present should be considered purely formal and given primarily for motivation.

## 数学代写|复分析作业代写Complex function代考|Elementary Properties of the Complex Numbers

$$(x, y)+\left(x^{\prime}, y^{\prime}\right)=\left(x+x^{\prime}, y+y^{\prime}\right),(x, y) \cdot\left(x^{\prime}, y^{\prime}\right) \quad=\left(x x^{\prime}-y y^{\prime}, x y^{\prime}+y x^{\prime}\right) .$$

$$\alpha \cdot(x, y)=(\alpha, 0) \cdot(x, y)=(\alpha x, \alpha y) .$$

$$(x+i y)+\left(x^{\prime}+i y^{\prime}\right)=\left(x+x^{\prime}\right)+i\left(y+y^{\prime}\right),(x+i y) \cdot\left(x^{\prime}+i y^{\prime}\right) \quad=\left(x x^{\prime}-y y^{\prime}\right)+i\left(x y^{\prime}+y x^{\prime}\right)$$

## 数学代写|复分析作业代写Complex function代考|Further Properties of the Complex Numbers

(1) 如果 $z=x$ 是真实的，那么
$$e^{z}=e^{x} \equiv \sum_{j=0}^{\infty} \frac{x^{j}}{j !}$$

(2) 如果 $z=i y$ 是纯虚数，那么
$$e^{z}=e^{i y} \equiv \cos y+i \sin y .$$
(3) 如果 $z=x+i y$ ，然后
$$e^{z}=e^{x+i y} \equiv e^{x} \cdot(\cos y+i \sin y) .$$

## 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|复变函数作业代写Complex function代考|Math 417

statistics-lab™ 为您的留学生涯保驾护航 在代写复变函数Complex function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。我们的专家在代写复变函数Complex function代写方面经验极为丰富，各种代写复变函数Complex function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|复变函数作业代写Complex function代考|The Influence of V. E. Katsnelson and D. Z. Arov on the Direction of Our Research Group

While working on generalized matricial Nehari problems (see $[21]$ ), Bernd and I made first contact with the works of V. E. Katsnelson. The problem is stated as follows:
GENERALIZED MATRICIAL NEHARI PROBLEM: Let $p, q \in \mathbb{N}$. Further, let $F_{11}$ and $F_{22}$ be a non-negative Hermitian $p \times p$ and a $q \times q$ measure, respectively, on the Borelian $\sigma$-Algebra $\mathfrak{B}{\mathbb{T}}$ on $\mathbb{T}:={z \in \mathbb{C}:|z|=1}$ and let $\left(\beta{k}\right){k=0}^{\infty}$ be a sequence of complex $p \times q$ matrices. Describe the set $\mathcal{F}\left(F{11}, F_{22},\left(\beta_{k}\right){k=0}^{\infty}\right)$ of all $\sigma$-additive mappings $F{12}$ from $\mathfrak{B}{\mathrm{T}}$ into the set of all complex $p \times q$ matrices fulfilling the conditions $$\int{\mathbb{T}} z^{-k} F_{12}(\mathrm{~d} z)=\beta_{k}, \quad k=0,1,2, \ldots$$
and for which
$$\left(\begin{array}{ll} F_{11} & F_{12} \ F_{12}^{} & F_{22} \end{array}\right)$$ is a non-negative Hermitian $(p+q) \times(p+q)$ measure on $\mathfrak{B}{\mathbb{T}}$. In particular, state necessary and sufficient conditions such that the set $\mathcal{F}\left(F{11}, F_{22},\left(\beta_{k}\right){k=0}^{\infty}\right)$ is non-empty. The problem stated above leads one to studying kernels on $\mathbb{N}{0} \times \mathbb{N}{0}$ of so-called mixed Toeplitz-Hankel type. To see this, for all $k \in \mathbb{Z}$, set $$\alpha{k}:=\int_{\mathbb{T}} z^{-k} F_{11}(\mathrm{~d} z) \quad \text { and } \quad \delta_{k}:=\int_{\mathbb{T}} z^{-k} F_{22}(\mathrm{~d} z)$$
and, for all $(m, n) \in \mathbb{N}{0} \times \mathbb{N}{0}$, define
$$K(m, n):=\left(\begin{array}{cc} \alpha_{m-n} & \beta_{m+n} \ \beta_{m+n}^{} & \delta_{n-m} \end{array}\right)$$
The kernel $K$ being non-negative definite turns out to be necessary and sufficient for the set $\mathcal{F}\left(F_{11}, F_{22},\left(\beta_{k}\right)_{k=0}^{\infty}\right)$ to be non-empty.

The just defined kernel $K$ is also important because of the following observation.
GENERALIZED HERGLOTZ-BOCHNER THEOREM: Let $p, q \in \mathbb{N}$ and let $\left(\alpha_{k}\right){k=0}^{\infty},\left(\beta{k}\right){k=0}^{\infty}$, and $\left(\delta{k}\right){k=0}^{\infty}$ be sequences belonging to $\mathbb{C}^{p \times p}, \mathbb{C}^{p \times q}$, and $\mathbb{C}^{q \times q}$, respectively. Then there exists a non-negative Hermitian $(p+q) \times(p+q)$ Borelian measure on T such that for all $m, n \in{0,1,2, \ldots}$ the equation $$K(m, n)=\int{\mathbb{T}}\left[\operatorname{diag}\left(z^{-m} I_{p}, z^{m} I_{q}\right)\right] F(\mathrm{~d} z)\left[\operatorname{diag}\left(z^{-n} I_{p}, z^{n} I_{q}\right)\right]^{*}$$
is satisfied if and only if $K$ is non-negative definite.

## 数学代写|复变函数作业代写Complex function代考|A Preamble

As a mathematician, Victor Katsnelson was raised within a fine school of function theory and functional analysis, which was blossoming in Kharkov starting the second half of 1930s. He studied in the Kharkov State University in 1960-1965. Among his teachers were Naum Akhiezer, Boris Levin, Vladimir Marchenko. That time he became acquainted with Vladimir Matsaev whom Victor often mentions as one of his teachers. In 1965 Katsnelson graduated with the master degree, Boris Levin supervised his master thesis. Since then and till 1990, he teaches at the Department of Mathematics and Mechanics of the Kharkov State University. In 1967 he defends the PhD Thesis “Convergence and Summability of Series in Root Vectors of Some Classes of Non-Selfadjoint Operators” also written under Boris Levin guidance. Until he left Kharkov in the early 1990s, Katsnelson remained an active participant of the Kharkov function theory seminar run on Thursdays by Boris Levin and Iossif Ostrovskii. His talks, remarks and questions were always interesting and witty.

Already in the 1960s Victor established himself among the colleagues as one of the finest Kharkov mathematicians of his generation, if not the finest one. Nevertheless, he was not appointed as a professor and was never allowed to travel abroad.

Most of Katsnelson’s work pertain to the spectral theory of functions and operators. I will touch only a handful of his results, mostly published in 1965-1970,that is, at the very beginning of his mathematical career. A big portion of his works written in Kharkov appeared in the local journal “Function Theory, Functional Analysis and Their Applications” and were never translated in English. Today, this journal is available at http://dspace.univer.kharkov.ua/handle/123456789/43.

In this occasion, let me mention two wonderful books carefully written by Katsnelson $[18,19]$. They exist only as manuscripts, and curiously, both have “Part I” in their titles, though, as far as I know, no continuations appeared. In both books mathematics interlaces with interesting historical comments. Last but not least, let me also mention an extensive survey of Issai Schur’s works in analysis written jointly by Dym and Katsnelson [7].

## 数学代写|复变函数作业代写Complex function代考|The Influence of V. E. Katsnelson and D. Z. Arov on the Direction of Our Research Group

GENERALIZED MATRICIAL NEHARI PROBLEM: Let $p, q \in \mathbb{N}$. 此外，让 $F_{11}$ 和 $F_{22}$ 是一个非负厄米特 $p \times p$ 和一个 $q \times q$ 分别测量 Borelian $\sigma$-代数 $\mathfrak{B T}$ 上T $:=z \in \mathbb{C}:|z|=1$ 然后让 $(\beta k) k=0^{\infty}$ 是一个昆杂的序列 $p \times q$ 矩阵。描述集合 $\mathcal{F}\left(F 11, F_{22},\left(\beta_{k}\right) k=0^{\infty}\right)$ 其中 $\sigma$ – 加法映射 $F 12$ 从 $\mathfrak{B}$ T 进入所有复数的集合 $p \times q$ 满足条件的矩阵
$$\int \mathbb{T} z^{-k} F_{12}(\mathrm{~d} z)=\beta_{k}, \quad k=0,1,2, \ldots$$

$$\left(\begin{array}{llll} F_{11} & F_{12} & F_{12} & F_{22} \end{array}\right)$$

$\mathcal{F}\left(F 11, F_{22},\left(\beta_{k}\right) k=0^{\infty}\right)$ 是非空的。上述问题导致研究内核 $\mathbb{N} 0 \times \mathbb{N} 0$ 所调的混合 Toeplitz-Hankel 类型。看到这 个，给大家 $k \in \mathbb{Z}$ ，放
$$\alpha k:=\int_{\mathbb{T}} z^{-k} F_{11}(\mathrm{~d} z) \quad \text { and } \quad \delta_{k}:=\int_{\mathbb{T}} z^{-k} F_{22}(\mathrm{~d} z)$$

$$K(m, n)=\int \mathbb{T}\left[\operatorname{diag}\left(z^{-m} I_{p}, z^{m} I_{q}\right)\right] F(\mathrm{~d} z)\left[\operatorname{diag}\left(z^{-n} I_{p}, z^{n} I_{q}\right)\right]^{*}$$

## 数学代写|复变函数作业代写Complex function代考|A Preamble

Katsnelson 的大部分工作都与函数和算子的谱理论有关。我将只触及他的一小部分结果，大部分发表于 1965-1970 年，也就是他数学生涯的初期。他用哈尔科夫写的大部分作品出现在当地期刊《泛函理论、泛函分析及其应用》上，从未翻译成英文。今天，该期刊可在 http://dspace.univer.kharkov.ua/handle/123456789/43 获得。

## 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|复变函数作业代写Complex function代考|AMATH 567

statistics-lab™ 为您的留学生涯保驾护航 在代写复变函数Complex function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。我们的专家在代写复变函数Complex function代写方面经验极为丰富，各种代写复变函数Complex function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|复变函数作业代写Complex function代考|Viktor Emmanuilovich’s Year as a Visiting Professor

In order to deepen the scientific collaboration with Viktor Emmanuilovich, Bernd and I intensively thought about possibilities to invite him to Leipzig for a longer period. This aim could indeed be pursued in 1991 on a large scale. At this point a few further words are advisable. One of the most famous scientists in history of Leipzig University is undoubtedly Wilhelm Ostwald (1853-1932), one of the founding fathers of physical chemistry, who was honored with the Nobel prize in chemistry in 1909. In honor of Wilhelm Ostwald, a chair named after him was established at Leipzig University, which was assigned to exceptionally renowned foreign guest researchers by the Faculty of Sciences of Leipzig University. Due to V. E.’s extensive publications in function theory and functional analysis, he had already gained a high reputation in the 1980 s. Among other things, this was particularly shown by an invitation to a week-long guest stay at the famous Weizmann Institute of Science Rehovot in summer 1990 . This sparked the idea to put forward the proposal to assign the Ostwald Chair to V. E. in the first half of 1991 . To our great delight, the proposal was accepted by the Faculty of Sciences of Leipzig University. From today’s perspective, Viktor Emmanuilovich turned out to be the last holder of the Ostwald Chair. The profound changes at Leipzig University after the political turnaround resulted in the abolishment of the Ostwald Chair for guest researchers. During the time of his visit to Leipzig, a big part of V. E.’s family emigrated to Israel. For this reason, he was unsure how and where he could continue his academic career. In order to support him, Bernd and I considered it advisable to extend his stay in Leipzig beyond the duration of the Ostwald Chair. The realization of this idea was complemented by a fortunate circumstance. The DFG offered multiple funding opportunities in order to support universities in the former GDR. Taking advantage of one of these programs, Bernd and I managed to arrange a DFG visiting professorship at Leipzig University for V. E. for the second half of 1991 . Eventually, he stayed in Leipzig for an entire year. On January 22,1992 , he then left Leipzig for the Weizmann Institute, where he was offered a professorship for Theoretical Mathematics (see Fig. 4).

## 数学代写|复变函数作业代写Complex function代考|The 60th Birthday of Viktor Emmanuilovich

In honor of V. E.’s 60 th birthday on September 3,2003 , we organized a workshop at Leipzig University which was among others attended by V. K. Dubovoy, W. Schempp, B. Silbermann. G. Heinig, A. Lasarow. Of course, a scientific contribution of the jubilarian was a must as well. He gave a two-part lecture about rational solutions of Schlesinger’s equation and their tau functions. At the end of the workshop on September 10,2003 , V. K. Dubovoy gave a very appropriate description of V. E. in form of an entry in the guest book of the Mathematical Institute of Leipzig University:
First and foremost, I would like to cordially thank Professor Bernd Kirstein and Professor Bernd Fritzsche for the invitation to Leipzig and the opportunity to speak at the conference in honor of the 60th birthday of Viktor Emmanuilovich Katsnelson.

I first encountered Viktor Emmanuilovich in spring 1963. Forty years have already passed since then. The predominant part of these years, I stood in close contact with Viktor Emmanuilovich. How many different topics were elucidated throughout!!! Regardless of a certain severity in his judgments, Viktor Emmanuilovich is very democratic company and also willing to expose himself to sharp criticism, which he did sufficiently often compared to others, too. To me, the contact with the mathematician Viktor Emmanuilovich was and allowed me to give up a whole series of illusions. I know Viktor Emmanuilovich as a person, who feels mathematics deeply and subtly and who strives to convey this feeling to others. He has written scientific papers that identify him as a great master.

You can compare Viktor Emmanuilovich to a singular point in our lives from which a mighty stream of energies emerges. It has not always been easy (just how it is not always easy for him), but without him the world would be poorer.

I express my wishes for Vitja through a passage from a poem by Boris Pasternak, which he loves a lot:
“… but be alive – this only matters – alive and burning to the end.”

## 数学代写|复变函数作业代写Complex function代考|The 60th Birthday of Viktor Emmanuilovich

“……但要活着——这很重要——活着并燃烧到最后。”

## 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|复变函数作业代写Complex function代考|Math 213A

statistics-lab™ 为您的留学生涯保驾护航 在代写复变函数Complex function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。我们的专家在代写复变函数Complex function代写方面经验极为丰富，各种代写复变函数Complex function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|复变函数作业代写Complex function代考|First Steps in Schur Analysis

After the defense of our joint dissertation on problems of the filter theory of multidimensional stationary sequences in December 1983, Bernd Fritzsche and I decided to aim our future research at the analytic foundation of prediction theory of multivariate stationary sequences. Against this background, we took up intense studies of the trend-setting works of the Soviet school (Kolmogorov, Rozanov, Matveev) as well as of American scholars (Wiener, Masani, Helson, Lowdenslager). During this process, we became aware of V. P. Potapov’s fundamental work [45] about the multiplicative structure of $J$-contractive matrix functions for the first time and we began to study the basics of $J$-theory systematically. Our choice of this research field was considerably encouraged by $P$. R. Masani. During Masanis’s visit of Leipzig University in May 1986 we had profound discussions about the state of prediction theory at that time and its prospects. P. R. Masani revealed to us that in collaboration with Norbert Wiener, following the works $[43,44,50,51]$, further research on an application of the results of V. P. Potapov in prediction theory was planned. However, the realization of this intention became unattainable due to Norbert Wiener’s death on March 18,1964 . Without Norbert Wiener P. R. Masani was reluctant to tackle this project and he turned towards a systematic elaboration of the theory of measures with orthogonal values in a Hilbert space or rather of the theory of orthoprojector-valued measures. P. R. Masani encouraged us to get in direct touch with the students of V. P. Potapov, who had passed away in the year 1980 . V. P. Potapov had been employed at the FTINT (Russian abbreviation for B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine in Kharkov) during the last period of his life (1976-1980) and he was able to contribute significantly to the popularization of $J$-Theory. In particular, he managed to assemble a group of exceedingly committed mathematicians in Kharkov who devoted themselves with inequalities to matricial versions of classical interpolation and moment problems. Among others, I. V. Kovalishina, V. E. Katsnelson, V. K. Dubovoy, L. B. Golinskii, I. V. Mikhailova, and Yu. M. Dyukarev belonged to this circle of mathematicians. In the summer of 1986 , Bernd Fritzsche and I decided to invite one representative of the circle of the above mentioned mathematicians to a month-long work visit at Leipzig University in the year \$1987.

## 数学代写|复变函数作业代写Complex function代考|Viktor Emmanuilovich’s First Visit to Leipzig

Over the course of planning the research activities for the year 1989 in fall 1988 , Bernd Fritzsche and I intended to hold a week-long international seminar on Schur analysis. This request was complied with by the administration of the Center for Theoretical Sciences (NTZ). The week from October 16-20, 1989 was specified as the date of the event. In connection with this seminar, we requested a 3-week stay for V. E. at Leipzig University. This request was granted as well. On September 23 , 1989 , he arrived in Leipzig. Aged 46 years it was his very first travel abroad. At the beginning of his stay it was unforeseeable that during his visit certain incidents would happen in Leipzig, which should stir up the political situation in the GDR significantly. V. E. became an eye-witness of the massive Monday protests in the city center of Leipzig on both October 9th and Uctober 16th that set the decay of the GDR in motion. Within a personal evaluation of these events, he reasoned that the reunification of Germany was the only consequence that seemed logical. We ourselves considered his prognosis very utopian at the time. History proved, though, that he had predicted everything completely correctly. Less than 1 year later, on October 3, 1990, the reunification of Germany was in fact enforced. During the first 3 weeks of his stay in Leipzig, Bernd and I had profound mathematical discussions with him, where he drew our attention to central problems in Schur analysis and, moreover, imparted to us fundamental aspects of the research of the Kharkov school. At that time, his lectures were held in Russian and I acted as interpreter into German for the audience (Fig. 2). At the end of his stay in Leipzig, the INTSEM (International Seminar) on Schur Analysis took place. It was P. R. Masani who had suggested the event during his first visit to Leipzig in 1986. The aim of this seminar was to gather leading specialists from the East and the West working on Schur analysis. This goal was successfully pursued. Among the Western participants were P. R. Masani, A. Dijksma, H. S. V. de Snoo, S. Hassi, and others. The list of Soviet participants included I. V. Kovalishina, V. E. Katsnelson, V. K. Dubovoy, Yu. L. Shmulyan, and I. M. Spitkovskii (Fig. 3). On October 17, 1989, the second day of the seminar, Mark Grigorevich Krein, one of the greatest mathematicians of the twentieth century, who had made fundamental contributions to Schur analysis and numerous other fields passed away. For this reason, D. Z. Arov has not been able to come to Leipzig for the seminar in time. He arrived on October 21 st, that is, 1 day after the end of the seminar. Following the seminar, it was intended that he would stay in Leipzig for another 3 weeks. During this time, the foundation for a long-term scientific collaboration with D. Z. Arov was set.

## 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。