Posted on Categories:Real analysis, 实分析, 数学代写

# 数学代写|实分析代写Real Analysis代考|Dirichlet’s Theorem

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|实分析代写Real Analysis代考|Dirichlet’s Theorem

Until now, we have not done full justice to Fourier’s series of trigonometric functions. In Chapter 1, we considered the expansion of an even function, a function for which $f(x)=f(-x)$. For such a function, we look for a cosine expansion
$$f(x)=a_0+a_1 \cos x+a_2 \cos 2 x+a_3 \cos 3 x+\cdots,$$
where $f$ is defined over the interval $(-\pi, \pi)$. There is an analogous sine series for any odd function, $g(x)=-g(-x)$ :
$$g(x)=b_1 \sin x+b_2 \sin 2 x+b_3 \sin 3 x+\cdots .$$
An arbitrary function can be expressed uniquely as the sum of an even function and an odd function (see exercises 6.1.4 and 6.1.5 at the end of this section). For an arbitrary function defined over the interval $(-\pi, \pi)$, we try to represent it as the sum of a cosine series and a sine series:
$$F(x)=a_0+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$
Fourier had considered such general series. The heuristic argument for finding the coefficients in an arbitrary Fourier series rests on the observation that for integer values of $k$ and $m$,
\begin{aligned} & \int_{-\pi}^\pi \cos (k x) \cos (m x) d x=\left{\begin{array}{cc} 0 & \text { if } \quad k \neq m, \ 2 \pi & \text { if } k=m=0, \ \pi & \text { if } k=m \neq 0, \end{array}\right. \ & \int_{-\pi}^\pi \sin (k x) \sin (m x) d x=\left{\begin{array}{cl} 0 & \text { if } k \neq m, \ \pi & \text { if } k=m \neq 0, \end{array}\right. \ & \int_{-\pi}^\pi \sin (k x) \cos (m x) d x=0 . \end{aligned}

## 数学代写|实分析代写Real Analysis代考|The Nature of the Problem

The first problem Fourier encountered was that of defining what he meant by
$$\int_{-\pi}^\pi F(x) \cos (k x) d x$$
In 1807, integration was defined and understood as the inverse process of differentiation, what some of today’s textbooks call “antidifferentiation.” The connection between integration and problems of areas and volumes was well understood, but that did not change the fact that one defined
$$\int F(x) \cos (k x) d x$$
as a function whose derivative was $F(x) \cos (k x)$.
This was a conceptual problem for many of those encountering Fourier series for the first time. It is not always true that $F(x) \cos (k x)$ can be expressed as the derivative of a known function. Fourier was responsible for changing the definition of an integral from an antiderivative to an area. It was his idea to put limits on the integration sign and to talk of a definite integral that was to be defined in terms of the area between $F(x) \cos (k x)$ and the $x$ axis.

Dirichlet was the first to realize that not every function could be integrated. He mentions the function that takes on one value at every rational number and a different value at every irrational number, for example
$$f(x)= \begin{cases}1, & \text { if } x \text { is rational } \ 0, & \text { if } x \text { is irrational }\end{cases}$$

## 数学代写|实分析代写Real Analysis代考|Dirichlet’s Theorem

$$f(x)=a_0+a_1 \cos x+a_2 \cos 2 x+a_3 \cos 3 x+\cdots,$$

$$g(x)=b_1 \sin x+b_2 \sin 2 x+b_3 \sin 3 x+\cdots$$

$$F(x)=a_0+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$

$\$ \\mid begin { aligned } \& \backslash int _{-}{-\backslash p i} \wedge|p i| \cos (k x) \mid \cos (m x) d x=\backslash left { 0 \quad if \quad k \neq m, 2 \pi \quad if k=m=0, \pi \quad if k=m \neq 0, |正确的。 । \& \backslash int _{-}{-\backslash p i} \wedge \backslash p i|\sin (k x)| \sin (m x) d x=\backslash left { 0 \quad if k \neq m, \pi \quad if k=m \neq 0, |正确的。। \& \mid int _{-}{-\mid p i} \wedge|p i| \sin (k x) \mid \cos (\mathrm{mx}) \mathrm{dx}=0 。 \结束 { 对齐 } \ \

## 数学代写|实分析代写Real Analysis代考|The Nature of the Problem

$$\int_{-\pi}^\pi F(x) \cos (k x) d x$$
1807 年，整合被定义并理解为分化的逆过程，今天的一些教科书称之为“反分化”。积分与面积和体 积问题之间的联系很好理解，但这并没有改变人们定义的事实
$$\int F(x) \cos (k x) d x$$

Dirichlet 是第一个意识到并非所有功能都可以集成的人。他提到函数在每个有理数处取一个值，在 每个无理数处取一个不同的值，例如
$$f(x)={1, \quad \text { if } x \text { is rational } 0, \quad \text { if } x \text { is irrational }$$

## 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。