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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

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

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

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

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法

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## 数学代写|数学物理方法代写Mathematical Methods代考|Bounded, unbounded, convergent, oscillatory

Bounded, unbounded, convergent, oscillatory. Let $M$ be an arbitrary positive number; it is possible that whatever $M$ we take there is at least one value of $8_{n}$ such that $\left|s_{n}\right|>M$. Such a sequence is called unbounded. $s_{n}=n$ is an obvious example, for we need only take $n$ to be any integer greater than $M$. By an argument similar to that for limit-points, an unbounded sequence must have an infinite number of terms such that $\left|s_{n}\right|$ is greater than any assigned $M$.

If we can choose an $M$ such that all $\left|s_{n}\right|$ are less than $M$, the sequence is called bounded. Both the sequences given at the end of $1.04$ are bounded; the condition holds for both if $M=3$.

If there is a number $s$ such that, given any positive number $\epsilon$, we can choose $m$ so that for every $n>m \quad\left|s_{n}-8\right|<\epsilon$, (1) the sequence is said to be convergent, and to have limit 8 . We then write* or $$\begin{gathered} 8_{n} \rightarrow 8 \quad(n \rightarrow \infty) \ \lim {n \rightarrow \infty} 8{n}=8 . \end{gathered}$$ The arrow is read ‘tends to’. We can write simply $$\lim s_{n}=8,$$ if no ambiguity is possible. Of the above examples $1 \cdot 04$ (1) is convergent with limit 0 ; we need only take $m>1 / \epsilon .1 \cdot 04(2)$ is not, because whatever $s$ and $m$ we take, if $\epsilon<\frac{1}{2}$, there will be terms with $n>m$ such that $\left|s_{n}-8\right| \geqslant \frac{1}{2}>\epsilon$.

The most important property of a convergent sequence is that if we have a rule for calculating each term, then we can calculate the limit to any accuracy we like. Some methods of approximation (cf. Chapters 9,17 ) will prove that a quantity lies within a given range, but this range is not arbitrarily small; the accuracy may be enough for the application in view but is not capable of being improved indefinitely.

A sequence that is bounded but not convergent is said to oscillate finitely, or simply to oscillate. An example is $1 \cdot 04(2)$; another is
$$s_{n}=(-1)^{n}+\frac{1}{n} .$$
Unlike 1.04 (2), all $s_{n}$ are different. The sequence is bounded, because $\left|s_{n}\right|<2$ for every $n$; but it does not converge since for large $n$ the members are alternately neer to 1 and $-1$, and (1) cannot be satisfied if $\epsilon<\frac{1}{2}$. If for any $M$ there is an $m$ such that $s_{n}>M$ for all $n>m$, we write
$s_{n}=n$ and $s_{n}=n^{2}$ are examples.
$$s_{n} \rightarrow \infty \text {. }$$
If for any $M$ there is an $m$ such that $s_{n}<-M$ for all $n>m$, we write
$s_{n}=-n$ and $s_{n}=-n^{2}$ are examples.
$$s_{n} \rightarrow-\infty \text {. }$$
Other types of unbounded sequences are represented by
$$s_{n}=(-1)^{n} n, \quad s_{n}=n \cos \frac{1}{2} \pi n, \quad s_{n}=n(1-\cos \pi n) .$$

## 数学代写|数学物理方法代写Mathematical Methods代考|Upper and lower bounds

Upper and lower bounds. A set (or sequence) bounded above has an upper bound; and one bounded below has a lower bound. The upper bound of a set is a quantity $M$ such that no member of the set exceeds $M$, but if $\epsilon$ is any positive quantity, however small, there is a member that exceeds $M-\epsilon$. The lower bound is a quantity $m$ such that no member is less than $m$, but there is always one less than $m+\epsilon$.

We use the method of Dedekind section. There are quantities $a$ such that $a$ is exceeded by some member of the set; for we might take an $a$ less than a known member of the set. Since the set is bounded above, there are quantities $b$ that are not exceeded by any member of the set. Every $b$ is greater than any $a$, and every quantity of the same dimensions is either an $a$ or a $b$. Hence the quantities $a$ form an $L$ and $b$ an $R$ class, and determine a cut, say at $M$. $M$ is a member of the $R$ class. For if it was a member of the $L$ class it would be exceeded by some member of the set, say $K$, and there would be no quantities $b$ between $M$ and $K$; hence $M$ would not be the quantity given by the cut. Hence no member of the set exceeds $M$. Also $M-\epsilon$ is in the $L$ class and therefore is exceeded by some member of the set. The corresponding result for lower bounds follows similarly.

The argument does not suppose the set infinite; but for a finite set the greatest of the set is the upper bound. For an infinite set all members may be less than the upper bound; for the set $1 \cdot 04$ (1) the upper bound is 1 and is equal to the first term, but the lower bound is 0 and no actual member is 0 .

What we call the upper bound is often called the least upper bound; and any quantity such that no member of the set exceeds it is then called an upper bound.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代想| Bounded, unbounded, convergent, oscillatory

(参见第9，17章) 将证明一个量位于给定的范围内，但这个范围不是任意小的;对于所查看的应用程序来说，准确性可能就足够

$$s_{n}=(-1)^{n}+\frac{1}{n} .$$

$$s_{n} \rightarrow \infty$$

$$s_{n} \rightarrow-\infty \text {. }$$

$$s_{n}=(-1)^{n} n, \quad s_{n}=n \cos \frac{1}{2} \pi n, \quad s_{n}=n(1-\cos \pi n)$$

## 数学代写|数学物理方法代写Mathematical Methods代想| Upper and lower bounds

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。

Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法

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## 数学代写|数学物理方法代写Mathematical Methods代考|Real numbers

Most of the present chapter will be already familiar to those who have studied a good modern book on calculus, and it is not intended to compete with standard works on pure mathematics. We think, however, that some discussion here is not out of place, for several reasons. First, the latter works for the most part do not emphasize why the refined arguments that they give have any relevance to physics, and physicists therefore tend to believe that they are irrelevant. Secondly, they are liable to be so long that a physicist can hardly be blamed if he decides that he has not the time to work through them. Thirdly, the attention to very peculiar functions has led the subject to be regarded as the pathology of functions. The reply is that every function, except an absolute constant, is peculiar somewhere, and that by studying where a function is peculiar we can arrive at constructive results about it that would be very hard to obtain otherwise. But we are entitled to regard ourselves as general practitioners and to restrict ourselves to the kinds of peculiarities that occur in physics; rare diseases may be handed over for treatment to a specialist, in this case a professional pure mathematician.

The nature of the problem was foreshadowed in a theorem of Euclid that the ratio of the hypotenuse to one side of an isosceles right-angled triangle is not equal to any rational fraction. Euclid, it must be remembered, made no use of what we should now call numerical measures of physical magnitudes. When he said that two lines were equal he meant that one could be placed on the other so that the two ends of one coincided with the two ends of the other; this is the direct physical comparison and does not require any numerical description of the lengths. When he said that the square on the hypotenuse was twice that on a side he meant that it could be cut into pieces and that the pieces could then be put together so as to make the square on the side twice over. He was working throughout with the quantities themselves, not with the numbers that we choose to associate with them in measurement with regard to any special unit. The use of numbers for this purpose is a choice of a language. What Euclid’s theorem showed was that the language of rational numbers was incapable of describing simultaneously the lengths of the side and the hypotenuse of a triangle that could easily be drawn by the rules of his geometry.

## 数学代写|数学物理方法代写Mathematical Methods代考|Nests of intervals: Dedekind section

Dedekind section. The fundamental property of real numbers is that they can be approximated to as closely as we please by rational numbers. When we say that
$$\sqrt{ } 2=1.414 \ldots,$$
we assert the following set of propositions: (1) 2 is between $1^{2}$ and $2^{2}$; (2) 2 is between $1 \cdot 4^{2}$ and $1 \cdot 5^{2}$; (3) 2 is between $1 \cdot 41^{2}$ and $1 \cdot 42^{2}$; (4) 2 is between $1 \cdot 414^{2}$ and $1 \cdot 415^{2}$; and so on to any desired accuracy. At each stage this process can be regarded as separating the decimals, to a given number of places, into two classes, those whose squares are respectively greater or less than 2 . At stage 3 , for instance, the squares of $1 \cdot 414,1 \cdot 413,1 \cdot 412$ are less than 2 , those of $1.415,1 \cdot 416,1.417$ greater than 2 . We say nothing at this stage about the fractions $1 \cdot 4141,1 \cdot 4142, \ldots, 1 \cdot 4149$; but at the next stage we say that 2 lies between the squares of $1 \cdot 4142$ and $1 \cdot 4143$. By taking a sufficient number of decimals we can make the unconsidered interval as small as we like, since we divide it by 10 at each step. Thus any decimal with a finite number of places will ultimately be classified according as its square is less or greater than 2. Now this process determines a unique infinite decimal, which we can take to be $\sqrt{ } 2$, and it can be regarded as the limit approached by the successive approximations from either side.

This process, which is capable of great extension, is an example of the definition of a

• Hence the name ‘irrational numbers’.

real number by a nest of rationals. We take a succession of rationals $\left{a_{n}\right}$ and another succession $\left{b_{n}\right}$, satisfying the following conditions:
(i) $a_{n+1} \geqslant a_{n}$
(ii) $b_{n+1} \leqslant b_{n}$,
(iii) $a_{n} \leqslant b_{n}$,
for all $n$, and
(iv) Given any positive rational number $\epsilon$, a number $N$ can be found such that $b_{n}-a_{n}<\epsilon$ for every $n>N$.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代想| Nests of intervals: Dedekind section

• 因此得名”无理数”。
实数由一窝有理数组成。我们采取一系列理性缺少或无法识别 \left 的分隔符 和另一个继承 缺少或无法识别 \left 的分隔符，满足以下条件:
(i) $a_{n+1} \geqslant a_{n}$
(二) $b_{n+1} \leqslant b_{n}$,
(三) $a_{n} \leqslant b_{n}$ ，
面向所有人 $n$, 和
(iv) 给定任何正有理数 $\epsilon_{r}$ 一个数字 $N$ 可以这样找到 $b_{n}-a_{n}<\epsilon$ 对于每个 $n>N$.

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。