Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|MATH2800 BACKGROUND CONCEPTS

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|ANALYTIC CONTINUATION

Let $f(z)$ be analytic in a given neighbourhood $N\left(z_{0} ; r_{0}\right)$ and $\Gamma$ be a closed Jordan curve starting and ending at $z_{0}$. Then we can find points $z_{0}, z_{1}, \ldots, z_{n}$ $=z_{0}$ on $\Gamma$ and neighbourhoods $N\left(z_{i}, r_{i}\right)\left(i=0,1, \ldots, n ; \dot{r}{n}=r{0}\right)$ such that $N\left(z_{i-1} ; r_{i-1}\right) \cap N\left(z_{i} ; r_{i}\right)(i=1,2, \ldots, n)$ are non-empty (Fig 1.1). Let $f_{1}(z)$ be analytic continuation of $f(z)$ into $N\left(z_{1} ; r_{1}\right), f_{2}(z)$ that of $f_{1}(z)$ into $N\left(z_{2} ; r_{2}\right)$ and so on, and finally let $f_{n}(z)$ be the continuation of $f_{n-1}(z)$ into $N\left(z_{n} ; r_{n}\right)$ $=N\left(z_{0} ; r_{0}\right)$, or $f_{n}(z)$ is the analytic continuation of $f(z)$ back into $N\left(z_{0} ; r_{0}\right)$ by means of the chain of analytic functions $f(z), f_{1}(z), \ldots, f_{n}(z)$. It can be shown that the function $f_{n}(z)$ is independent of the set of points $z_{0}, z_{1}, \ldots, z_{n}$ $=z_{0}$ chosen on $\Gamma$ and that the same function $f_{n}(z)$ is obtained if $\Gamma$ is replaced by any other such closed cur re passing through $z_{0}$ which can be continuously deformed to $\Gamma$ without crossing any singularity of the complete analytic function of which $f(z)$ is an element. Hence writing $\varphi(z)$ $=f_{n}(z)$, we say that the analytic function $f(z)$ defined in $N\left(z_{0} ; r_{0}\right)$ changes to $\varphi(z)$ by analytic continuation after description of $\Gamma$. Clearly, then the analytic function $\varphi(z)$ in $N\left(z_{0} ; r_{0}\right)$ changes to $f(z)$ after description of $\Gamma$ in the opposite sense.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|MULTIPLE-VALUED FUNCTIONS

The multiple-valuedness of complex functions arises from that of $\arg {z}$ which these functions invariably involve. As we know $\arg {z}$ is defined to be a real number such that $z=|z| \exp (i \arg {z})(z \neq 0)$ so that if $\theta$ is a value of $\arg {z}$, then $\theta+2 k \pi(k=\pm 1, \pm 2, \ldots)$ are also possible values of $\arg {z}$; the principal value of $\arg {z}$, to be denoted by $\operatorname{Arg}{z}$, is taken to be that value of $\arg {z}$ which lies in the interval $(-\pi, \pi]$ so that the different values of $\arg {z}$ are $\operatorname{Arg}{z}+2 k \pi(k=0, \pm 1, \pm, \ldots)$.
Now the multiple values of $\arg {z}$ at all points of the complex plane, except $z=0$, may be sorted into a number of branch functions $f_{k}(z)=$ $\operatorname{Arg}{z}+2 k \pi(k=0, \pm 1, \pm 2, \ldots)$, each of which is defined in the punctured plane $\mathbf{C}-{0}$ and is continuous therein except the negative real axis which is a line of jump discontinuity. If $\mathbf{C}^{}$ denotes the complex plane obtained by introducing a cut along the negative real axis, then each function $f_{k}(z)$ is continuous in $\mathbf{C}^{}$.

The function $\log z$ defined as $\log z=\log |z|+\arg {z}(z \neq 0)$ is evidently multiple-valued. The different branches of this function are $\log z+2 k \pi i(k=0, \pm 1, \pm 2, \ldots)$ where $\log z=\log |z|+i \operatorname{Arg}{z}$ is the principal branch; each branch is defined in $\mathbf{C}-{0}$ and is analytic in the cut-plane $\mathbf{C}^{}$. If $\alpha$ is any complex constant, $\alpha$ th power of $z$ defined to be $\exp (\alpha \log z)$ $(z \neq 0)$ is a multiple-valued function whose different branches are $\exp (\alpha \log z) e^{2 k \pi \alpha i}=z^{\alpha} e^{2 k \pi \alpha i}(k=0, \pm 1, \pm 2, \ldots)$ where $z^{\alpha}=\exp (\alpha \log z)$ is the principal branch; each branch is defined in $\mathbf{C}-{0}$ and is analytic in the cut-plane $\mathrm{C}^{}$. The multiple-valued function representing the $\alpha$ th power of $z$ will also be often denoted by $z^{\alpha}$ in the absence of any other suitable notation.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|ANALYTIC CONTINUATION

1.1) 。让 $f_{1}(z)$ 是分析的延续 $f(z)$ 进入 $N\left(z_{1} ; r_{1}\right), f_{2}(z)$ 那个 $f_{1}(z)$ 进入 $N\left(z_{2} ; r_{2}\right)$ 以此类推，最后㱼 $f_{n}(z)$ 成为的延续 $f_{n-1}(z)$ 进 $\lambda N\left(z_{n} ; r_{n}\right)=N\left(z_{0} ; r_{0}\right)$ ，或者 $f_{n}(z)$ 是分析的延续 $f(z)$ 回到 $N\left(z_{0} ; r_{0}\right)$ 通过分析函数琏 $f(z), f_{1}(z), \ldots, f_{n}(z)$. 可以证明函 数 $f_{n}(z)$ 独立于点集 $z_{0}, z_{1}, \ldots, z_{n}=z_{0}$ 选择在 $\Gamma$ 和相同的功能 $f_{n}(z)$ 获得如果「被任何其他这样的闭合电流通过 $z_{0}$ 可以连续弯形为 $\Gamma$ 不湂越其完整解析函数的任何奇点 $f(z)$ 是一个元青。因此写作 $\varphi(z)=f_{n}(z)$, 我们说解析函数 $f(z)$ 定义在 $N\left(z_{0} ; r_{0}\right)$ 更改为 $\varphi(z)$ 通过在䅦述后的分析延续 $\Gamma$. 显然，解析函数 $\varphi(z)$ 在 $N\left(z_{0} ; r_{0}\right)$ 更改为 $f(z)$ 描述后 $\Gamma$ 在相反的意义上。

## 数学代写|常微分方程代考Ordinary Differential Equations代写|MULTIPLEVALUED FUNCTIONS

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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。