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# 数学代写|数学物理方法代写Mathematical Methods代考|MAP6506 Double series

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## 数学代写|数学物理方法代写Mathematical Methods代考|Double series

Similar remarks apply to double series, in which the general term is $u_{m, n}$. The condition of convergence is now that we can choose $m, n$ so that for all $p$ greater than $m$ and all $q$ greater than $n$, the sums $\sum_{r=1}^{p} \sum_{s=1}^{Q} u_{r, s}, \sum_{r=1}^{m} \sum_{s=1}^{n} u_{r, 8}$ differ by a quantity with modulus less than $\epsilon$. Absolute and conditional convergence can be defined similarly, and it is again true that an absolutely convergent double series has the same sum however the terms are arranged. The proofs differ only in complexity from those for simple series.

## 数学代写|数学物理方法代写Mathematical Methods代考|Limits of functions: Continuity

In the most general sense, when we say that $f(x)$ is a function of $x$ in some range of values of $x$ we mean that for every value of $x$ in the range one or more values of $f(x)$ exist. We can, for instance, speak of a function of $x$ that is equal to 1 if $x$ is rational but to 0 if $x$ is irrational. Such a function would be fairly regarded by a physicist as pathological, and he is interested in a much narrower class of functions, roughly speaking such as can be represented by graphs.* It will usually also be required that the function shall be single-valued, but not necessarily. Thus for the circle
$$x^{2}+y^{2}=a^{2},$$
we have
$$y=\pm \sqrt{ }\left(a^{2}-x^{2}\right) \text {, }$$
and $y$ is a function of $x$; but we get its values over the whole circle only by taking both signs for the root. A single-valued function of $x$ in a range is one that has precisely one value for each value of $x$. We shall in the first place consider single-valued functions only.
The essential idea of a limit of a function is similar to that of the convergence of a sequence; for the terms of a sequence $\left{s_{n}\right}$ are the values of a function of the positive integral variable $n$, which is permitted to take arbitrarily large values. The new feature is that for a function $f(x)$ the variable $x$ is not restricted to be integral; it may be permitted to take any value over an interval or even any value however large.

When the values of $x$ form an interval we can define a limit of $f(\xi)$ as $\xi \rightarrow x$ as follows: if there is a quantity $c$ such that given any positive $\epsilon$ there is a positive $\delta$ such that whenever $0<|\xi-x|<\delta$, then $|f(\xi)-c|<\epsilon$, we say that $c$ is the limit of $f(\xi)$ as $\xi \rightarrow x$. (We may further restrict the admissible values of $\xi$ and, for instance, speak of the limit of $f(\xi)$ as $\xi-x \rightarrow 0$ through positive or negative values.) If also $c=f(x)$, we say that $f(\xi)$ is continuous at $\xi=x$. Then the definition of continuity may be stated as follows: if for any positive $\epsilon$ we can choose a positive $\delta$ such that whenever $|h|<\delta$
$$|f(a+h)-f(a)|<\epsilon$$
then $f(x)$ is said to be continuous at $x=a$. If this condition is satisfied and we take any sequence $\left{h_{n}\right}$ tending to 0 , then for any $\delta$ there will be an $m$ such that $\left|h_{n}\right|<\delta$ for all $n \geqslant m$, and then $\left|f\left(a+h_{n}\right)-f(a)\right|<\epsilon$. Hence for all such sequences $f\left(a+h_{n}\right) \rightarrow f(a)$.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代想| Limits of functions:Continuity

$$x^{2}+y^{2}=a^{2},$$

$$y=\pm \sqrt{ }\left(a^{2}-x^{2}\right),$$

$$|f(a+h)-f(a)|<\epsilon$$

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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。