statistics-labTM为您提供澳大利亚国立大学（The Australian National UniversityAdvanced Complex Analysis高级复分析澳洲代写代考辅导服务！

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Complex differentiability, conformal mapping; complex integration, Cauchy integral theorems, Taylor series representation, isolated singularities, residue theorem and applications to real integration. Topics chosen from: argument principle, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.

Note: This is an HPC. It emphasises mathematical rigour and proof and develops the material from an abstract viewpoint.

Consider
\begin{aligned} & f(z)=\frac{x y^2(x+\mathrm{i} y)}{x^2+y^4} \quad(z=x+\mathrm{i} y \neq 0) \ & f(0)=0 \end{aligned}
Verify that $\lim _{z \rightarrow z_0}\left(f(z)-z_0\right) /\left(z-z_0\right)=0$ as $z \rightarrow 0$ along any straight line, $z=(a+\mathrm{i} b) t, t \in \mathbb{R}$. This does not prove that $f^{\prime}(0)=0$, however. By considering $z \rightarrow 0$ along the path $z(t)=t^2+\mathrm{i} t$, show that $f$ is not differentiable at 0 . (This shows that when computing $f^{\prime}$ it is not enough to consider a limit taken along certain specific paths or types of path. An entire neighbourhood of the point concerned must be considered.)

Prove that if each of the series $\sum a_n z^n, \sum b_n z^n$, and $\sum a_n b_n z^n$ has radius of convergence equal to 1 , then so have the series $\sum a_n b_n^2 z^n$ and $\sum a_n^2 b_n z^n$.

For $\left|a_n\right| \leq 1$, show that $\sum a_n z^n$ is absolute convergent for all $|z|<1$. If $\sum a_n z^n=$ $f(z)$ for $\left|a_n\right|<1,|z|<1$, show that
$$|f(z)| \leq \frac{1}{1-|z|}$$

Prove that, for $z \neq 1$,
$$\sum_{n=1}^k \frac{z^n}{n}=\frac{z}{1-z}\left(\sum_{n=1}^{k-1} \frac{1}{n(n+1)}-\sum_{n=1}^{k-1} \frac{z^n}{n(n+1)}+\frac{1-z^k}{k}\right)$$
Show that the series $\sum_{n=1}^{\infty} z^n / n$ and $\sum_{n=1}^{\infty} z^n /(n(n+1))$ have radius of convergence 1 ; that the latter series converges everywhere on $|z|=1$, while the former series converges everywhere on $|z|=1$ except $z=1$.

Suppose that the power series $\sum_{n=0}^{\infty} a_n z^n$ has a recurring sequence of coefficients; that is, $a_{n+k}=a_n$ for some fixed positive integer $k$ and all $n$. Prove that the series converges for $|z|<1$ to a rational function $p(z) / q(z)$ where $p, q$ are polynomials, and that the roots of $q$ are all on the unit circle.
What happens if $a_{n+k}=a_n / k$ instead?

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。