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# 数学代写|数学物理方法代写Mathematical Methods代考|MATH210 Functions oftwo variables

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## 数学代写|数学物理方法代写Mathematical Methods代考|Functions oftwo variables

1-11. Functions of two variables. So far we have been considering sequences, which may be regarded as functions of one variable capable of taking only integral values, and functions of a continuous variable. In what follows we shall be concerned with what are essentially functions of two variables, which may be either integral or continuous. This introduces new complications when limiting processes are used, since it is not always obvious, or even true, that the same result will be obtained when the order of the limiting processes is changed. The simplest sufficient condition for the reversibility of limiting processes is provided by the following theorem on absolute convergence.
1-111. If $f(x, y)$ is a non-decreasing function of both $x$ and $y$ (either or both of which may take only integral values), and
$$\lim {x \rightarrow \infty} f(x, y)=g(y), \quad \lim {y \rightarrow \infty} f(x, y)=h(x),$$
then
$$\lim {y \rightarrow \infty} g(y)=\lim {x \rightarrow \infty} h(x),$$
in the sense that if either of the limits in (2) exists the other exists and the two are equal. Note first that $g(y)$ is a non-decreasing function of $y$. For if $y_2>y_1$
$$g\left(y_2\right)-g\left(y_1\right)=\lim _{x \rightarrow \infty}\left{f\left(x, y_2\right)-f\left(x, y_1\right)\right} \geqslant 0$$

## 数学代写|数学物理方法代写Mathematical Methods代考|Uniform convergence of sequences and series

1-112. Uniform convergence of sequences and series. The terms of a sequence $\left{f_n(x)\right}$ may be functions of a variable $x$. Then if the sequence converges for all values of $x$ in an interval, its limit is a function of $x$, say $f(x)$. If we choose an arbitrarily small positive $\epsilon$ we shall for any $x$ be able to choose $n(x)$ so that $\left|f_p(x)-f(x)\right|<\epsilon$ for all $p \geqslant n(x)$ because the sequence converges. In general the least value of $n(x)$ such that this is true will depend on $x$. But it may be possible to choose an $n$ independent of $x$ such that $\left|f_p(x)-f(x)\right|<\epsilon$ for all $p>n$ and for all $x$ in the interval. If this is possible for every $\epsilon, f_n(x)$ is said to be uniformly convergent to $f(x)$ in the interval. It can fail to be uniformly convergent if there is an $x$, say $c$, within or at the end of the interval such that if we take a succession of values of $x$, tending to $c$, the corresponding values of $n(x)$ for given $\epsilon$ tend to infinity.
As $f_n(x)$ may be the sum of the first $n$ terms of a series, all these statements have immediate analogues for series $\Sigma u_n(x)$ over an interval of $x$. Thus the series $\Sigma x^n$ converges for all $x$ such that $0 \leqslant x<1$, but it is not uniformly convergent for all such $x$. For if we fix $\epsilon$ and then choose $n$ so as to make $$x^n+x^{n+1}+\ldots+x^{n+p}=\frac{x^n\left(1-x^{p+1}\right)}{1-x}$$ less than $\epsilon$ for all $p \geqslant 1$ we must make $$x^n<(1-x) \epsilon,$$ and therefore $$n>\frac{\log {(1-x) \epsilon}}{\log x},$$
which tends to infinity as $x$ tends to 1 . This series is therefore uniformly convergent in a range $a \leqslant x \leqslant b$, where $a$ and $b$ are fixed quantities between 0 and $+1$, since we can choose $n$ greater than the greater of the quantities
$$\frac{\log {(1-a) \epsilon}}{\log a}, \frac{\log {(1-b) \epsilon}}{\log b},$$
and the same value of $n$ will then do for any intermediate value of $x$. It is convergent for any $x$ such that $-1<x<1$. But it is not uniformly convergent in the range $-1<x<1$ because, even though the signs < exclude the possibilities that $x$ may be actually $-1$ or $+1$, they permit any intermediate value, however close to 1 , and however we choose $n$ we shall always be able to find values of $x$ such that (3) is false.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代考|Functions oftwo variables

1-11。两个变量的函数。到目前为止，我们一直在考㨿序列，它可以被看作是一个只能取整数值的变量的函数，以及一个连紶变量 的函数。在下文中，我们将关殶什么是两个变量的本质函数，它们可以是积分的也可以是连续的。这在使用限制过程时引入了新的 复杂性，因为当改变限制过程的顺暄获得相同的结果并不总是显而易见的，甚至不是真的。极限过程可逆性的最简单充分条件由 以下关于绝对收敛的定理是供。
1-111。如果 $f(x, y)$ 是两者的非道咸函数 $x$ 和 $y$ (其中一个或两个只能取整数值)，以及
$$\lim x \rightarrow \infty f(x, y)=g(y), \quad \lim y \rightarrow \infty f(x, y)=h(x),$$

$$\lim y \rightarrow \infty g(y)=\lim x \rightarrow \infty h(x),$$

〈left 缺少或无法识别的分隔符

## 数学代写数学物理方法代写Mathematical Methods代考|Uniform convergence of sequences and series

1-112。序列和级数的一致收敛。序列的术语\left 缺少或无法识别的分隔符

$$\frac{\log (1-a) \epsilon}{\log a}, \frac{\log (1-b) \epsilon}{\log b}$$

（3）是错误的。

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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。