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

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

The expression $x+\mathrm{i} y$ for a complex number is intimately related to Cartesian coordinates $(x, y)$ in the plane. It turns out often to be useful to work with polar coordinates $(r, \theta)$, which we recall correspond to a point distance $r$ from the origin making an angle $\theta$ measured from the positive $x$-axis in an anticlockwise direction, Figure 1.5. Of course we measure $\theta$ in radians. These coordinate systems are related as follows:
\begin{aligned} & x=r \cos \theta \ & y=r \sin \theta \end{aligned}

Therefore
$$r=\sqrt{x^2+y^2}=|z|$$
where $z=x+\mathrm{i} y$.
Finding $\theta$ is slightly trickier because it is not unique. Any value of $\theta$ for which (1.19) holds is called an argument of $z$. The article ‘an’ is used to reflect the lack of uniqueness: if $\theta$ is an argument then so is $\theta+2 k \pi$ for any integer $k$. With the understanding that $\theta$ is unique only up to multiples of $2 \pi$, we may use the notation
$$\theta=\arg z$$
Often the choice of $\theta$ is rendered unique by imposing some convention: for example, we may insist that $\theta$ is chosen in the interval $[0,2 \pi)$, or in $(-\pi, \pi]$. The unique value of $\theta$ in the interval $(-\pi, \pi]$ is known as the principal value of the argument. (We follow standard practice in taking this particular interval. Its main advantage is that $\theta$ then behaves nicely near the positive real axis, where $\theta=0$. But this is a technical point that only acquires importance much later. The non-uniqueness of $\theta$ is a phenomenon with tremendous ramifications in the theory, as we shall see.

## 数学代写|复分析作业代写Complex function代考|The Complex Numbers Cannot be Ordered

The real numbers may be given an ordering (the usual one, $>$ ) which has among its properties the following:
If $x \neq 0$ then either $x>0$ or $-x>0$, but not both
If $x, y>0$ then $x+y>0, x y>0$

No such ordering can be defined on the complex numbers. Suppose for a contradiction that one can. Since $\mathrm{i} \neq 0,(1.20)$ implies that either $\mathrm{i}>0$ or $-\mathrm{i}>0$. Then (1.21) implies that either $-1=\mathrm{i} \cdot \mathrm{i}>0$ or $-1=(-\mathrm{i}) \cdot(-\mathrm{i})>0$. At the same time, $1=(-1)^2>0$. But then both 1 and -1 are greater than 0 , contrary to (1.20).

It is therefore not possible to use inequalities, analogous to those for reals, when discussing complex numbers. Any inequality that occurs must involve only real numbers, possibly related to the given complex numbers. For example, if $z \in \mathbb{C}$ then
$$z>1$$
makes no sense, but either of
$$|z|>1$$
or
$$\operatorname{re}(z)>1$$
is acceptable. (They do not mean the same thing!) As a convention, if we write a statement such as
$$\varepsilon>0$$
this will automatically imply that $\varepsilon$ is assumed to be a real number.

# 复分析代写

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

\begin{aligned} & x=r \cos \theta \ & y=r \sin \theta \end{aligned}

$$r=\sqrt{x^2+y^2}=|z|$$

$$\theta=\arg z$$

## 数学代写|复分析作业代写Complex function代考|The Complex Numbers Cannot be Ordered

$$z>1$$

$$|z|>1$$

$$\operatorname{re}(z)>1$$

$$\varepsilon>0$$

## 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|Overview of the Book

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## 数学代写|复分析作业代写Complex function代考|Overview of the Book

It is often useful to set the development of a mathematical theory in its historical context, but it is not always necessary to fight the historical battles again. In this text we give honour where we can to those pioneers who carved their way through uncharted mathematical territory. But more recent developments let us see the theory itself in a new light. To the modern ear the very name ‘complex analysis’ carries misleading overtones: it suggests complexity in the sense of complication. The older meaning, ‘composite’, was perhaps appropriate when the ‘real part’ of a complex number had a quite different status from that of the ‘imaginary part’. But nowadays a complex number is a perfectly integrated whole. To think of complex analysis as if it were, so to speak, two copies of real analysis, is to place undue emphasis on the algebra at the expense of the geometry, which in the long run has been far more influential. And in fact complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.

In preparing our approach to the subject we have adopted two basic organising principles. The first is the direct generalisation of real analysis to the complex case. Definitions, of limits, continuity, differentiation, and integration are natural extensions of the corresponding real notions. Since nowadays any student taking a course in complex analysis may be assumed to have made a study of the real counterpart, many battles have already been won. We can refer students to their accumulated knowledge, pausing only to phrase it appropriately. This saves time and energy, allowing us to proceed straight to the heart of the subject, where the interesting differences occur. Invariably this happens because the plane has a richer geometry than the line, and this leads to our second major organising principle: geometric insight is valuable and should be cultivated. Of course this insight must be translated into sound formal arguments; this can often be done using modern topological notions.

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

We begin with the definition that emerged from the insights of Wallis, Wessel, Argand, Gauss, and Hamilton:

DEFINITION 1.1. A complex number is an ordered pair $(x, y)$ of real numbers. Addition and multiplication of complex numbers are defined by:
\begin{aligned} \left(x_1, y_1\right)+\left(x_2, y_2\right) & =\left(x_1+x_2, y_1+y_2\right) \ \left(x_1, y_1\right)\left(x_2, y_2\right) & =\left(x_1 x_2-y_1 y_2, x_1 y_2+x_2 y_1\right) \end{aligned}
For example,
$$(3,5)(2,7)=(3 \cdot 2-5 \cdot 7,3 \cdot 7+5 \cdot 2)=(-29,31)$$

This definition is the culmination of several centuries of struggle to understand complex numbers, and it shows how elusive a simple idea can be. Before we see what these pairs have to do with $\sqrt{-1}$, however, let us establish some of their properties.

THEOREM 1.2. The set of complex numbers, with the operations defined by (1.1, 1.2), is a field. That is, the following axioms hold: if $z_1=\left(x_1, y_1\right), z_2=\left(x_2, y_2\right)$, and $z_3=$ $\left(x_3, y_3\right)$ are complex numbers, then
(a) Addition and multiplication are commutative:
\begin{aligned} z_1+z_2 & =z_2+z_1 \ z_1 z_2 & =z_2 z_1 \end{aligned}
(b) Addition and multiplication are associative:
\begin{aligned} \left(z_1+z_2\right)+z_3 & =z_1+\left(z_2+z_3\right) \ \left(z_1 z_2\right) z_3 & =z_1\left(z_2 z_3\right) \end{aligned}
(c) There is an additive identity $(0,0)$ :
$$z_1+(0,0)=z_1$$
(d) There is a multiplicative identity $(1,0)$ :
$$z_1(1,0)=z_1$$
(e) Each element has an additive inverse:
$$(x, y)+(-x,-y)=(0,0)$$
(f) Each element other than $(0,0)$ has a multiplicative inverse:
$$(x, y)\left(\frac{x}{x^2+y^2}, \frac{-y}{x^2+y^2}\right)=(1,0)$$
$$z_1\left(z_2+z_3\right)=z_1 z_2+z_1 z_3$$

# 复分析代写

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

1.1.定义复数是一对有序的实数$(x, y)$。复数的加法和乘法定义为:
\begin{aligned} \left(x_1, y_1\right)+\left(x_2, y_2\right) & =\left(x_1+x_2, y_1+y_2\right) \ \left(x_1, y_1\right)\left(x_2, y_2\right) & =\left(x_1 x_2-y_1 y_2, x_1 y_2+x_2 y_1\right) \end{aligned}

$$(3,5)(2,7)=(3 \cdot 2-5 \cdot 7,3 \cdot 7+5 \cdot 2)=(-29,31)$$

## 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。