### 数学代写|微积分代写Calculus代写|MATH7000

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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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|微积分代写Calculus代写|Power Series and the Extension Theorem

An analytic function is one that can be expanded as a convergent power series
$$f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^{n}$$
in some open disk $S={z:|z-a|<r}$ centered at $a$ with radius $r$. The term “analytic” describes this property over a set: an analytic function on a given domain can be represented as a convergent power series at any point in the domain. You recognize such a power series from your studies of Taylor series $f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^{n}$ in realvalued calculus (where the variable $z$ was real). Just as the root (or ratio) test from realvariable calculus can determine the (positive) radius of convergence $R$ (which provides the power series’ largest such open interval of convergence ${z=x:|x-a|<R$ for $x \in \mathbb{R}}$ ), so does the test work for complex functions to provide the largest such open disk of convergence ${z:|z-a|<R$ for $z \in C}$.

One of the most important and beautiful theorems of complex analysis is that “functions holomorphic at all points in a disk are analytic over that disk!” Combined with the fact that analytic functions are holomorphic, we see that differentiability (on a disk) is exactly equivalent to the expression of a complex function as a convergent power series. Differentiability of the analytic function is assured throughout the disk of convergence. Hence analytic exactly means holomorphic at all points on a disk. This fact has many beautiful and powerful positive consequences, first realized by AugustinLouis Cauchy in [31]. He developed the formula for an analytic function’s Taylor series based on an integral identity. Our pathway toward that important formula will use as stepping stones the following four foundational theorems.

## 数学代写|微积分代写Calculus代写|The Exponential Function and Polar Coordinates

This section already mentioned the idea contained in the Extension Theorem: the important functions arising as analytic functions in real-valued calculus have natural extensions to complex functions. The calculus properties learned in real-valued calculus have beautiful and natural extensions out to the complex functions’ extensions. Before proving the theorems, this subsection shows how exponential functions produce a superb illustration of their use. How does the much-beloved real-valued function $f(x)=\mathrm{e}^{x}$ extend into the complex plane? In other words, how must we properly define a complex function $f(z)=\mathrm{e}^{z}$ so that it agrees with $f(x)=\mathrm{e}^{x}$ for all real $z=x$ ? The Extension Theorem says the answer must be equivalent to the one found by simply replacing $x$ with $z$ in the familiar real power series
$$\mathrm{e}^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$
Since this real series has an infinite radius of convergence, so must the complex series when we replace $x$ with $z$, and then it will correctly define $\mathrm{e}^{z}$ for all $z \in \mathrm{C}$. Furthermore, the expression
$$\mathrm{e}^{a+b}=\mathrm{e}^{u} \mathrm{e}^{b}$$
is exactly equivalent to the formal power series expression
$$\sum_{n=0}^{\infty} \frac{(a+b)^{n}}{n !}=\left(\sum_{n=0}^{\infty} \frac{a^{n}}{n !}\right)\left(\sum_{n=0}^{\infty} \frac{b^{n}}{n !}\right)$$
Because these expressions hold for real $a$ and $b$ and can both be formulated in terms of power series, the Extension Theorem (as these series always converge) implies they both hold for any complex $a$ and $b$. Then the real power series generates, via the Extension Theorem, the complex function $\mathrm{e}^{z}=\mathrm{e}^{x+i y}$ in the following way:
\begin{aligned} \mathrm{e}^{z} &=\sum_{n=0}^{\infty} \frac{z^{n}}{n !}=\sum_{n=0}^{\infty} \frac{(x+i y)^{n}}{n !}=\left(\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\right)\left(\sum_{n=0}^{\infty} \frac{(i y)^{n}}{n !}\right) \ &=\mathrm{e}^{x} \sum_{n=0}^{\infty} \frac{i^{n} y^{n}}{n !}=\mathrm{e}^{x}\left(\sum_{n=0}^{\infty} \frac{(-1)^{n} y^{2 n}}{(2 n) !}+i \sum_{n=0}^{\infty}(-1)^{n} \frac{y^{2 n+1}}{(2 n+1) !}\right) \end{aligned}

## 数学代写|微积分代写Calculus代写|Power Series and the Extension Theorem

$$f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^{n}$$

$f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^{n}$ 在实值微积分中 (其中变量 $z$ 是真实的)。正如实变量微积分的根 (或比率) 检验 可以确定 (正) 收敛半径 $R$ (它提供了幂级数最大的这种开收敛区间 $z=x:|x-a|<R \$$for \ x \in \mathbb{R} )，对 于复杂函数的测试工作也是如此，以提供最大的这种开放式收敛圆盘 z:|z-a|<R \$$ for$\$z \in C$.

## 数学代写|微积分代写Calculus代写|The Exponential Function and Polar Coordinates

$$\mathrm{e}^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$

$$\mathbf{e}^{a+b}=\mathbf{e}^{u} \mathbf{e}^{b}$$

$$\sum_{n=0}^{\infty} \frac{(a+b)^{n}}{n !}=\left(\sum_{n=0}^{\infty} \frac{a^{n}}{n !}\right)\left(\sum_{n=0}^{\infty} \frac{b^{n}}{n !}\right)$$

$$\mathrm{e}^{z}=\sum_{n=0}^{\infty} \frac{z^{n}}{n !}=\sum_{n=0}^{\infty} \frac{(x+i y)^{n}}{n !}=\left(\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\right)\left(\sum_{n=0}^{\infty} \frac{(i y)^{n}}{n !}\right) \quad=\mathrm{e}^{x} \sum_{n=0}^{\infty} \frac{i^{n} y^{n}}{n !}=\mathrm{e}^{x}\left(\sum_{n=0}^{\infty} \frac{(-1)^{n} y^{2 n}}{(2 n) !}\right.$$

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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。