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# 数学代写|实分析代写Real Analysis代考|The Riemann Integral

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## 数学代写|实分析代写Real Analysis代考|The Riemann Integral

A more useful definition of integration was given by Bernhard Riemann in “Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe.” As mentioned in section 5.1 , this was written after the summer of 1852 when Riemann had discussed questions of Fourier series with Dirichlet. Its purpose was nothing less than to find necessary and sufficient conditions for a function to have a representation as a trigonometric series. Riemann never published it, probably because it raised many new questions that he was hoping to answer. It appeared in 1867 , after his death.

Riemann begins with a summary of the history of the subject, describing the contributions from d’Alembert, Euler, Bernoulli, and Lagrange and the questions that arose concerning the validity of a trigonometric expansion for arbitrary functions. He discusses Fourier’s contributions and Dirichlet’s proof, emphasizing Dirichlet’s recognition of the distinction between absolute and conditional convergence. This is where the Riemann rearrangement theorem is stated, not as a theorem but as an observation. He points out the difficulty with Fourier series: that in general the convergence will not be absolute.

This is followed by a list of the assumptions that Dirichlet needed to impose on a function in order to prove that it did have representation as a trigonometric series:
I. it must be integrable,
II. at each point of discontinuity, its value must be the average of the limit from the left and the limit from the right,
III. it must be piecewise continuous, bounded, and piecewise monotonic.
The second condition is essential. We have seen that the Fourier series cannot equal the original function at any point where this is not true. The third assumption is not as clearly necessary. Most of Riemann’s work involved probing how far the third assumption could be weakened.

## 数学代写|实分析代写Real Analysis代考|The Riemann Integral

The first task is to clarify the meaning of the integral. Cauchy’s definition was adequate for proving that any bounded continuous function is integrable. It is also sufficient for a demonstration that any bounded piecewise continuous function is integrable. Riemann wished to consider even more general functions, functions with infinitely many discontinuities within any finite interval. His definition is very similar to Cauchy’s. Like Cauchy,

he uses approximating sums:
$$\int_a^b f(x) d t \approx \sum_{j=1}^n f\left(x_{j-1}^\right)\left(x_j-x_{j-1}\right) .$$ Unlike Cauchy who evaluated the function $f$ at the left-hand endpoint of each interval, Riemann allows approximating sums in which $x_{j-1}^$ can be any point in the interval $\left[x_{j-1}, x_j\right]$. Because of this extra freedom, it appears more difficult to guarantee convergence of these series. In fact, for bounded functions Riemann’s definition is equivalent to Cauchy’s. Cauchy wanted to be able to prove that any continuous function is integrable. Riemann was interested in seeing how discontinuous a function could be and still remain integrable. As he realized, to be tied to the left-hand endpoints obscures what is happening in general. Riemann’s definition-in the language of the $\epsilon-\delta$ game-is the following.

Definition: integration (Riemann)
A function $f$ is said to be Riemann integrable over the interval $[a, b]$ and its integral has the value $V$ provided that the following condition is satisfied. Given any specified error bound $\epsilon$, there must be a response $\delta$ such that for any partition
$$a=x_0<x_1<x_2<\cdots<x_n=b$$
where each of the subintervals has length less than $\delta$,
$$\left|x_j-x_{j-1}\right|<\delta, \quad \text { for all } j \text {, }$$
and for any set of values $x_0^* \in\left[x_0, x_1\right], x_1^* \in\left[x_1, x_2\right], \ldots, x_{n-1}^* \in\left[x_{n-1}, x_n\right]$, the corresponding approximating sum will lie within $\epsilon$ of the value $V$,
$$\left|\sum_{j=1}^n f\left(x_{j-1}^*\right)\left(x_j-x_{j-1}\right)-V\right|<\epsilon .$$
The value of the integral is denoted by
$$V=\int_a^b f(x) d x$$

## 数学代写|实分析代写Real Analysis代考|The Riemann Integral

Bernhard Riemann 在“Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe” 中给出了更有用的积分定义。如 5.1 节所述，这是在 1852 年夏天黎埐与狄利克雷讨论傅里叶级数问题之后写 的。它的目的不亚于寻找一个函数具有三角级数表示的充分必要条件。黎妟从末发表过它，可能是因为它提出 了许多他㸴望回答的新问题。它出现于 1867 年，在他死后。

II. 在每个不连续点，其值必须是左侧极限和右侧极限的平均值，
III。它必须是分段连续的、有界的和分段单调的。

## 数学代写|实分析代写Real Analysis代考|The Riemann Integral

$$a=x_0<x_1<x_2<\cdots<x_n=b$$

$$\left|x_j-x_{j-1}\right|<\delta, \quad \text { for all } j$$

$$\left|\sum_{j=1}^n f\left(x_{j-1}^*\right)\left(x_j-x_{j-1}\right)-V\right|<\epsilon .$$

$$V=\int_a^b f(x) d x$$

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