### 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|why study complex analysis

These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis. First, it is, in my humble opinion, one of the most beautiful areas of mathematics. One way of putting it that has occurred to me is that complex analysis has a very high ratio of theorems to definitions (i.e., a very low “entropy”): you get a lot more as “output” than you put in as “input.”

The second reason is complex analysis has a large number of applications (in both the pure math and applied math senses of the word) to things that seem like they ought to have little to do with complex numbers. For example:

• Solving polynomial equations: historically, this was the motivation for introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543 , after learning of the solution found earlier by Scipione del Ferro. An important point to keep in mind is that Cardano’s formula sometimes requires taking operations in the complex plane as an intermediate step to get to the final answer, even when the cubic equation being solved has only real roots.

Example 1. Using Cardano’s formula, it can be found that the solutions to the cubic equation
$$z^3+6 z^2+9 z+3=0$$
are
\begin{aligned} & z_1=2 \cos (2 \pi / 9)-2, \ & z_2=2 \cos (8 \pi / 9)-2, \ & z_3=2 \sin (\pi / 18)-2 \end{aligned}

• Proving Stirling’s formula: $n ! \sim \sqrt{2 \pi n}(n / e)^n$. Here, $a_n \sim b_n$ is the standard “asymptotic to” relation, defined to mean $\lim _{n \rightarrow \infty} a_n / b_n=1$.
• Proving the prime number theorem: $\pi(n) \sim \frac{n}{\log n}$, where $\pi(n)$ denotes the number of primes less than or equal to $n$ (the prime-counting function).

## 数学代写|复分析作业代写Complex function代考|The fundamental theorem of algebra

One of the most famous theorems in complex analysis is the not-very-aptly named Fundamental Theorem of Algebra. This seems like a fitting place to start our journey into the theory.

Theorem 1 (The Fundamental Theorem of Algebra.). Every nonconstant polynomial $p(z)$ over the complex numbers has a root.

The fundamental theorem of algebra is a subtle result that has many beautiful proofs. I will show you three of them. Let me know if you see any “algebra”…
First proof: analytic proof. Let
$$p(z)=a_n z^n+a_{n-1} z^{n-1}+\ldots+a_0$$
be a polynomial of degree $n \geq 1$, and consider where $|p(z)|$ attains its infimum.
First, note that it can’t happen as $|z| \rightarrow \infty$, since
$$|p(z)|=|z|^n \cdot\left(\left|a_n+a_{n-1} z^{-1}+a_{n-2} z^{-2}+\ldots+a_0 z^{-n}\right|\right)$$
and in particular $\lim {|z| \rightarrow \infty} \frac{|p(z)|}{|z|^n}=\left|a_n\right|$, so for large $|z|$ it is guaranteed that $|p(z)| \geq|p(0)|=\left|a_0\right|$. Fixing some radius $R>0$ for which $|z|>R$ implies $|p(z)| \geq\left|a_0\right|$, we therefore have that $$m_0:=\inf {z \in \mathbb{C}}|p(z)|=\inf {|z| \leq R}|p(z)|=\min {|z| \leq R}|p(z)|=\left|p\left(z_0\right)\right|$$
where $z_0=\underset{|z| \leq R}{\arg \min }|p(z)|$, and the minimum exists because $p(z)$ is a continuous function on the disc $D_R(0)$.

Denote $w_0=p\left(z_0\right)$, so that $m_0=\left|w_0\right|$. We now claim that $m_0=0$. Assume by contradiction that it doesn’t, and examine the local behavior of $p(z)$ around $z_0$; more precisely, expanding $p(z)$ in powers of $z-z_0$ we can write
$$p(z)-w_0+\sum_{j=1}^n c_j\left(z-z_0\right)^j-w_0+c_k\left(z-z_0\right)^k+\ldots+c_n\left(z-z_0\right)^n$$

where $k$ is the minimal positive index for which $c_j \neq 0$. (Exercise: why can we expand $p(z)$ in this way?) Now imagine starting with $z=z_0$ and traveling away from $z_0$ in some direction $e^{i \theta}$. What happens to $p(z)$ ? Well, the expansion gives
$$p\left(z_0+r e^{i \theta}\right)=w_0+c_k r^k e^{i k \theta}+c_{k+1} r^{k+1} e^{i(k+1) \theta}+\ldots+c_n r^n e^{i n \theta}$$

## 数学代写|复分析作业代写Complex function代考|why study complex analysis

• 求解多项式方程: 从历史上看，这是卡尔达诺引入复数的动机，卡尔达诺在学习了 Scipione del Ferro 较早发现的解后，于 1543 年发表了蓍名的求解三次方程的公式。需要牢记的重要一点是，卡 尔达诺公式有时需要在复平面上进行运算作为获得最终答案的中间步骤，即使所求解的三次方程只有 实根也是如此。
例 1. 利用卡尔达诺公式，可以求出三次方程的解
$$z^3+6 z^2+9 z+3=0$$

$$z_1=2 \cos (2 \pi / 9)-2, \quad z_2=2 \cos (8 \pi / 9)-2, z_3=2 \sin (\pi / 18)-2$$
• 证明斯特林公式: $n$ ! $\sqrt{2 \pi n}(n / e)^n$. 这里， $a_n \sim b_n$ 是标准的“渐近”关系，定义为 $\lim _{n \rightarrow \infty} a_n / b_n=1$.
• 证明素数定理: $\pi(n) \sim \frac{n}{\log n}$ ，在哪里 $\pi(n)$ 表示小于等于的素数个数 $n$ (质数计数函数）。

## 数学代写|复分析作业代写Complex function代考|The fundamental theorem of algebra

$$p(z)=a_n z^n+a_{n-1} z^{n-1}+\ldots+a_0$$

$$|p(z)|=|z|^n \cdot\left(\left|a_n+a_{n-1} z^{-1}+a_{n-2} z^{-2}+\ldots+a_0 z^{-n}\right|\right)$$

$$m_0:=\inf z \in \mathbb{C}|p(z)|=\inf |z| \leq R|p(z)|=\min |z| \leq R|p(z)|=\left|p\left(z_0\right)\right|$$

$$p(z)-w_0+\sum_{j=1}^n c_j\left(z-z_0\right)^j-w_0+c_k\left(z-z_0\right)^k+\ldots+c_n\left(z-z_0\right)^n$$

$$p\left(z_0+r e^{i \theta}\right)=w_0+c_k r^k e^{i k \theta}+c_{k+1} r^{k+1} e^{i(k+1) \theta}+\ldots+c_n r^n e^{i n \theta}$$

## 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。