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## 数学代写|复分析作业代写Complex function代考|The Zeros of a Holomorphic Function

Let $f$ be a holomorphic function. If $f$ is not identically zero, then it turns out that $f$ cannot vanish at too many points. This once again bears out the dictum that holomorphic functions are a lot like polynomials. The idea has a precise formulation as follows:

Theorem 3.6.1. Let $U \subseteq \mathbb{C}$ be a connected open set and let $f: U \rightarrow \mathbb{C}$ be holomorphic. Let $\mathbf{Z}={z \in U: f(z)=0}$. If there are a $z_{0} \in \mathbf{Z}$ and $\left{z_{j}\right}_{j=1}^{\infty} \subseteq \mathbf{Z} \backslash\left{z_{0}\right}$ such that $z_{j} \rightarrow z_{0}$, then $f \equiv 0$.

Let us formulate Theorem 3.6.1 in topological terms. We recall that a point $z_{0}$ is said to be an accumulation point of a set $Z$ if there is a sequence $\left{z_{j}\right} \subseteq Z \backslash\left{z_{0}\right}$ with $\lim {j \rightarrow \infty} z{j}=z_{0}$. Then Theorem 3.6.1 is equivalent to the statement: If $f: U \rightarrow \mathbb{C}$ is a holomorphic function on a connected open set $U$ and if $Z={z \in U: f(z)=0}$ has an accumulation point in $U$, then $f \equiv 0$.

There is still more terminology attached to the situation in Theorem 3.6.1. A set $S$ is said to be discrete if for each $s \in S$ there is an $\epsilon>0$ such that $D(s, \epsilon) \cap S={s}$. People also say, in an abuse of language, that a discrete set has points which are “isolated” or that $S$ contains only “isolated points.” Theorem 3.6.1 thus asserts that if $f$ is a nonconstant holomorphic function on a connected open set, then its zero set is discrete or, less formally, the zeros of $f$ are isolated. It is important to realize that Theorem 3.6.1 does not rule out the possibility that the zero set of $f$ can have accumulation points in $\mathbb{C} \backslash U$; in particular, a nonconstant holomorphic function on an open set $U$ can indeed have zeros accumulating at a point of $\partial U$. For example, the function $f(z)=\sin (1 /(1-z))$ is holomorphic on $U=D(0,1)$ and vanishes on the set
$$\mathbf{Z}=\left{1-\frac{1}{\pi n}: n=1,2,3, \ldots\right} .$$

## 数学代写|复分析作业代写Complex function代考|The Behavior of a Holomorphic Function

In the proof of the Cauchy integral formula in Section 2.4, we saw that it is often important to consider a function that is holomorphic on a punctured open set $U \backslash{P} \subset \mathbb{C}$. The consideration of a holomorphic function with such an “isolated singularity” turns out to occupy a central position in much of the subject. These singularities can arise in various ways. Perhaps the most obvious way occurs as the reciprocal of a holomorphic function, for instance passing from $z^{j}$ to $1 / z^{j}, j$ a positive integer. More complicated examples can be generated, for instance, by exponentiating the reciprocals of holomorphic functions: for example, $e^{1 / z}, z \neq 0$.

In this chapter we shall study carefully the behavior of holomorphic functions near a singularity. In particular, we shall obtain a new kind of infinite series expansion which generalizes the idea of the power series expansion of a holomorphic function about a (nonsingular) point. We shall in the process completely classify the behavior of holomorphic functions near an isolated singular point.

Let $U \subseteq \mathbb{C}$ be an open set and $P \in U$. Suppose that $f: U \backslash{P} \rightarrow \mathbb{C}$ is holomorphic. In this situation we say that $f$ has an isolated singular point (or isolated singularity) at $P$. The implication of the phrase is usually just that $f$ is defined and holomorphic on some such “deleted neighborhood” of $P$. The specification of the set $U$ is of secondary interest; we wish to consider the behavior of $f$ “near $P$ “.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|The Zeros of a Holomorphic Function

left 的分隔符缺失或无法识别

$\lim j \rightarrow \infty z j=z_{0}$. 那么定理 3.6.1 等价于陈述: 如果 $f: U \rightarrow \mathbb{C}$ 是连通开集上的全纯函数 $U$ 而如果 $Z=z \in U: f(z)=0$ 有一个积男点 $U$ ，然后 $f \equiv 0$.

$f(z)=\sin (1 /(1-z))$ 是全纯的 $U=D(0,1)$ 然后消失在片场

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## MATLAB代写

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

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