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## 数学代写|微积分代写Calculus代写|Laurent Series

Just as a Taylor series provides important information about a function’s behavior, such as an ability to approximate the function by just a few terms of the Taylor series, a function’s Laurent series provides important information about functions that have isolated singularities. Just as a Taylor series representation is valid over a disk of convergence, so is a Laurent series valid over a particular region called an annulus of convergence. This section describes what these Laurent series look like, formally states and proves the fact that functions with isolated singularities have such a series, and shows how to determine a function’s Laurent series and its annulus of convergence when the expansion is about a given isolated singularity.

An annulus is a ring-shaped object bounded by two concentric circles. See Figure 3.3. An annulus can always be described algebraically as
$$\left{z: r<\left|z-z_{0}\right|<R\right},$$
where $r$ and $R$ are nonnegative extended real numbers. (The radius $r$ can equal 0 and $R$ can equal $\infty$.) These shapes form analytic domains that turn out to be required for a function to have a Laurent series representation. Conversely, when a function is analytic uver such an anmular domain, then it always has a Laurent series representation. What is this Laurent series, and why is it useful? This is the question explored in this section.

## 数学代写|微积分代写Calculus代写|Connection to Fourier Series.

When the Laurent series is taken about $z_{0}=0$ and the contour is the unit circle
$$\partial \mathbb{D}=\left{z: z=\mathrm{e}^{i t} \text { for }-\pi \leq t<\pi\right},$$
Laurent’s Theorem gives a series representation $f(z)=\sum_{n=-\infty}^{\infty} c_{n} z^{n}$ for any function analytic over any annulus centered at 0 and that contains $\partial \mathbb{D}$. Parameterize the coefficient’s integral formula
$$c_{n}=(2 \pi i)^{-1} \oint_{C} f(z)(z-0)^{-(n+1)} d z$$
given in Laurent’s Theorem, noting
$$z^{-(n+1)} d z=i \mathrm{e}^{-i n t} d t \quad \text { for }-\pi \leq t<\pi .$$
The result is:
$$f\left(\mathrm{e}^{i t}\right)=\sum_{n=-\infty}^{\infty} c_{n} \mathrm{e}^{i n t}, \quad \text { where } c_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f\left(\mathrm{e}^{i t}\right) \mathrm{e}^{-i n t} d t$$
In a similar way, mathematicians often want to write a real-valued function $f(t)$ as
$$f(t)=\sum_{n=-\infty}^{\infty} c_{n} \mathrm{e}^{i n t} \quad \text { and } \quad c_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) \mathrm{e}^{-i n t} d t$$
This is called the Fourier series representation for $f$. You can see that in both formats $f$ is thought of as a function of the real variable $t$ on the interval $[-\pi, \pi)$. The two expressions-the Laurent series and the Fourier series-look the same, so long as the composition with the exponential in the Laurent series expression is suppressed, reformatting $f\left(\mathrm{e}^{i t}\right)$ with a renaming $f(t)$.

But a sleight of hand has happened here, and we have to take care in what we believe! The two expressions are of the same format but are different and are applied in different scenarios. For the renaming has actually been a change of variables. We can be more careful and process it correctly. Write $f\left(\mathrm{e}^{i t}\right)=f(x)$ or, equivalently, $x=\mathrm{e}^{i t}$. Then as a single-valued function, $t=-i \log x$, which is not differentiable over the unit circle. (It is discontinuous at $-1$.) Writing $f(t)$ is actually writing $f(-i \log x)$, and the hypotheses of Laurent’s Theorem may not be in play for this function of $x$. But instead of holding back, mathematicians proceed boldly: they formulate the Fourier series in a formal way for almost any real-valued function. But the question remains about when the series $\sum_{n=-\infty}^{\infty} c_{n} \mathrm{e}^{i n x}$ converges and when it converges to $f(x)$. That was a major research question throughoul the 1800 s and well into the twentieth century.

## 数学代写|微积分代写Calculus代写|Laurent Series

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## 数学代写|微积分代写Calculus代写|Connection to Fourier Series.

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$$c_{n}=(2 \pi i)^{-1} \oint_{C} f(z)(z-0)^{-(n+1)} d z$$

$$z^{-(n+1)} d z=i \mathrm{e}^{-i n t} d t \quad \text { for }-\pi \leq t<\pi$$

$$f\left(\mathrm{e}^{i t}\right)=\sum_{n=-\infty}^{\infty} c_{n} \mathrm{e}^{i n t}, \quad \text { where } c_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f\left(\mathrm{e}^{i t}\right) \mathrm{e}^{-i n t} d t$$

$$f(t)=\sum_{n=-\infty}^{\infty} c_{n} \mathrm{e}^{i n t} \quad \text { and } \quad c_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) \mathrm{e}^{-i n t} d t$$

## 有限元方法代写

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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。

assignmentutor™您的专属作业导师
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