### 数学代写|微积分代写Calculus代写|MTH2010

statistics-lab™ 为您的留学生涯保驾护航 在代写微积分Calculus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。我们的专家在代写微积分Calculus代写方面经验极为丰富，各种代写微积分Calculus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|微积分代写Calculus代写|Multivalued Functions and Riemann Surfaces

Just as for real functions, a complex function $f^{-1}$ is the inverse function of a complex function $f$ when
$f^{-1}(f(z))=z$ for all $z$ in the domain of $f$
and
$w=f\left(f^{-1}(w)\right)$ for all $w$ in the range of $f$
The inverse is under composition (and is not the multiplicative inverse). Inverse functions are important because their application solves $f(w)=z$ for $w$. In other words, they provide the algebraic formula to solve important equations. For example, $f(z)=$ $z^{2}$ has inverse $f^{-1}(z)=\sqrt{z}$, which we can properly define. It solves $w^{2}=z$ for $w$, since (applying the inverse function to both sides) it means $w=\sqrt{z}$. The square root function is certainly not the only inverse function of interest. Likewise, $f^{-1}(z)=\sqrt[3]{z}$ solves $z=w^{3}$. Similarly, $f^{-1}(z)=z^{5 / 4}$ solves $z=w^{4 / 5}$. In the same way, we would like to properly define the logarithm function $f^{-1}(z)=\log z$ to solve $z=\mathrm{e}^{w}$. Each of these inverse function definitions follow the same structural pathway, and this section generally describes how that works.

The square root function provides a good place to start. The Extension Theorem’s demand that the square root function is an extension of its real-valued counterpart implies $\sqrt{z}=\sqrt{x}$ for real $z=x \geq 0$, but how is $f^{-1}(z)=\sqrt{z}$ defined for other complex values? It turns out, as we see below, that the polar representation of the domain value $z$ (and then the straightforward analysis of how the algebra of the inverse function should work) produces the correct definition of $f^{-1}(z)$.

## 数学代写|微积分代写Calculus代写|Riemann Surfaces

Since a multivalued function takes on more than one output value $f(z)$ at a given $z$, the branches’ output values $f_{1}(z), f_{2}(z)$, etc. will be range sets in the complex plane $\mathbb{C}$. They will each have the same domain. Bernhard Riemann had the clever idea of “gluing” the domain sets together, one on top of the other, to produce a single domain embedded in three dimensions.

The effect is to produce a single “sheet” in 3-space (with the complex plane as its two horizontal dimensions) that makes up the “total domain” of the multivalued function. Such a surface is called a Riemann surface, and the individual pieces of the sheet that correspond to each branch of the function is called a branch of the multivalued function’s domain. There would then be, for example, infinitely many domain sheet pieces above a given value of $z$ when $f$ has an infinite number of branches there. Furthermore, Riemann realized it could be possible to “glue” together the domain pieces to produce two very satisfying results. First, mathematicians can consider the function as defined on the Riemann surface’s domain, and then it is not multivalued. Instead, it maps each domain point on the Riemann surface to exactly one range value in $\mathbb{C}$. Second, this resulting function $f(z)$ could be considered as continuous over its Riemann surface, whereas the multivalued function isn’t continuous across any branch cut. In other words, when complex points $z$ and $w$ are close together in the Riemann surface, the output values $f(z)$ and $f(w)$ can be close together in the range space. In short, the inherently pleasant analytic properties of the function can exist when considering the Riemann surface as the domain, whereas the branch cuts destroy the multivalued function’s property of continuity. Nice!

Here’s the key to doing this successfully: Examine the range output for each branch. “Cut” each branch’s domain (in $\mathbb{C}$ ) along a slice that corresponds to where the branch’s range output has a boundary. This is called a branch cut. Each domain is a flat two-dimensional region. For each one, “pull” one side of the branch cut up into 3-space and the other side down. Then “glue” together the domain pieces of those branches so that the output points glue together, too, forming a continuous map on the Riemann surface domain.

## 数学代写|微积分代写Calculus代写|Multivalued Functions and Riemann Surfaces

$f^{-1}(f(z))=z$ 对所有人 $z$ 在领域 $f$

$w=f\left(f^{-1}(w)\right)$ 对所有人 $w$ 在范围内 $f$

$f^{-1}(z)=\sqrt[3]{z}$ 解决 $z=w^{3}$. 相似地， $f^{-1}(z)=z^{5 / 4}$ 解决 $z=w^{4 / 5}$. 同样，我们想正确定义对数函数
$f^{-1}(z)=\log z$ 解决 $z=\mathrm{e}^{w}$. 这些反函数定义中的每一个都遵循相同的结构路径，本节一般描述其工作原理。

## 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。