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# 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier-Bessel Series

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier-Bessel Series

BESSEL FUNCTIONS ARISE IN MANY PROBLEMS in physics possessing cylindrical symmetry, such as the vibrations of circular drumheads and the radial modes in optical fibers. They also provide us with another orthogonal set of basis functions.
Bessel functions have a long history and were named after Friedrich Wilhelm Bessel (1784-1846).
The first occurrence of Bessel functions (zeroth order) was in the work of Daniel Bernoulli on heavy chains (1738). More general Bessel functions were studied by Leonhard Euler in 1781 and in his study of the vibrating membrane in 1764. Joseph Fourier found them in the study of heat conduction in solid cylinders and Siméon Poisson (1781-1840) in heat conduction of spheres (1823).

The history of Bessel functions did not just originate in the study of the wave and heat equations. These solutions originally came up in the study of the Kepler problem, describing planetary motion. According to G. N. Watson in his Treatise on Bessel Functions, the formulation and solution of Kepler’s Problem was discovered by JosephLouis Lagrange (1736-1813), in 1770. Namely, the problem was to express the radial coordinate and what is called the “eccentric anomaly,” $E$, as functions of time. Lagrange found expressions for the coefficients in the expansions of $r$ and $E$ in trigonometric functions of time. However, he only computed the first few coefficients. In 1816, Friedrich Wilhelm Bessel (1784-1846) had shown that the coefficients in the expansion for $r$ could be given an integral representation. In 1824 , he presented a thorough study of these functions, which are now called “Bessel functions.”

You might have seen Bessel functions in a course on differential equations as solutions of the differential equation
$$\mathrm{x} 2 y^{\prime \prime}+\mathrm{xy} y^{\prime}+(\mathrm{x} 2-\mathrm{p} 2) \mathrm{y}=0 .(3.82)$$
Solutions to this equation are obtained in the form of series expansions. Namely, one seeks solutions of the form
$$y(x)=\Sigma j=0 \text { ooaj } j+n$$
by determining the form the coefficients must take. We will leave this for a homework exercise and simply report the results.

## 数学代写|傅里叶分析代写Fourier Analysis代考|The Least Squares Approximation

We want to measure the deviation of the finite sum from the given function. Essentially, we want to look at the error made in the approximation. This is done by introducing the mean square deviation:
$$\mathrm{EN}=\int \mathrm{ab}[\mathrm{f}(\mathrm{x})-\mathrm{SN}(x)] 2 \rho(x) \mathrm{dx},$$
where we have introduced the weight function $\rho(x)>0$. It gives us a sense as to how close the $N$ th partial sum is to $f(x)$.

We want to minimize this deviation by choosing the right $c_n \mathrm{~s}$. We begin by inserting the partial sums and expand the square in the integrand:
$$\begin{array}{r} E N=\int a b[f(x)-S N(x)] 2 \rho(x) d x=\int a b\left[f(x)-\sum n=1 N c n \phi n(x)\right] 2 \rho(x) d x=\int a b f 2(x) \rho(x) d x \ -2 \int a b f(x) \Sigma n=1 N c n \phi n(x) \rho(x) d x+\int a b \Sigma n=1 N c n \phi n(x) \Sigma m=1 N c m \phi m(x) \rho(x) d x . \end{array}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier-Bessel Series


\ mathrm {x} 2 y ^ {\ ‘ \ ‘} + \ mathrm y ^ {xy} {\ ‘} + (\ mathrm {x} 2 – \ mathrm {p} 2) \ mathrm {y} = 0。(3.82)



y(x)=\Sigma j=0 \text {ooaj} j+n


## 数学代写|傅里叶分析代写Fourier Analysis代考|The Least Squares Approximation


int \ mathrm {EN} = \ \ mathrm {ab} [\ mathrm {f} (\ mathrm {x}) – \ mathrm {SN} (x)) 2 \ρ(x) \ mathrm {dx},



r \开始{数组}{}
E N=\int a b[f(x)- s N(x)] 2 \rho(x) d x=\int a b\left[f(x)-\sum N= 1 N c N \ N \ N(x) \右]2 \rho(x) d x=\int a b f 2(x) \rho(x) d x \
-2 \int a b f(x) \Sigma n=1 n c n \ n(x) \rho(x) d x+\int a b \Sigma n=1 n c n \ n(x) \Sigma m=1 n cm \ m(x) \rho(x) d x。



## 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。