Riemann integral

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Analysis Ⅲ 2020 Oxford PDF

Contents

Preface

Chapter 1. Step functions and the Riemann integral
1.1. Step functions
1.2. $I$ of a step function
1.3. Definition of the integral
1.4. Basic theorems about the integral
1.5. Simple examples
1.6. Not all functions are integrable

Chapter 2. Basic theorems about the integral
2.1. Continuous functions are integrable
2.2. Mean value theorems
2.3. Monotone functions are integrable

Chapter 3. Riemann sums

Chapter 4. Integration and differentiation
4.1. First fundamental theorem of calculus
4.2. Second fundamental theorem of calculus
4.3. Integration by parts
4.4. Substitution

Chapter 5. Limits and the integral
5.1. Interchanging the order of limits and integration
5.2. Interchanging the order of limits and differentiation
5.3. Power series and radius of convergence

Chapter 6. The exponential and logarithm functions
6.1. The exponential function
6.2. The logarithm function

Chapter 7. Improper integrals

Preface

These are the notes for Analysis III at Oxford. The objective of this course is to present a rigorous theory of what it means to integrate a function $f:[a, b] \rightarrow \mathbb{R}$. For which functions $f$ can we do this, and what properties does the integral have? Can we give rigorous and general versions of facts you learned in school, such as integration by parts, integration by substitution, and the fact that the integral of $f^{\prime}$ is just $f$ ?

We will present the theory of the Riemann integral, although the way we will develop it is much closer to what is known as the Darboux integral. The end product is the same (the Riemann integral and the Darboux integral are equivalent) but the Darboux development tends to be easier to understand and handle.

This is not the only way to define the integral. In fact, it has certain deficiencies when it comes to the interplay between integration and limits, for example. To handle these situations one needs the Lebesgue integral, which is discussed in a future course.

Students should be aware that every time we write "integrable" we mean "Riemann integrable". For example, later on we will exhibit a non-integrable function, but it turns out that this function is integrable in the sense of Lebesgue.

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翻页时MathJax.typeset()与TikZ

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Chapter 4(第5页)的svg图是用MetaPost画的. 备份一下代码:

1. u = 10cm; v = 20cm;
2. xmax = .5; ymax = .275;
3. drawarrow (-xmax*u, 0) -- (xmax*u, 0); label.bot(btex $x$ etex, (xmax*u, 0));
4. drawarrow (0, -ymax*v) -- (0, ymax*v); label.lft(btex $y$ etex, (0, ymax*v));
5. numeric pi; pi := 3.1415926;numeric radian; radian := 180/pi;
6. vardef sin primary i = (sind(i*radian)) enddef;
7. path curve;curve:=(0,0)
8. for x=.05 step .005 until .5:
9. -- (x,x**2 * sin(1/x**2))xscaled u yscaled v
10. endfor;
11. draw curve;
12. draw curve rotated 180;

复制代码

MetaPost 在线预览

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对于 一元 实值 函数 的 Riemann 积分 没有 采用 通常的 达布上和、下和 的 定义 方法，而是 用 阶梯 函数 逼近 的 方法 ， 用 阶梯 函数 序列 对 定义 函数 的 Riemann 可积性 及 其 Riemann 积分。这样 处理 有 一个 好处，当 把 绝对值 换成 一般 范数 时 ， 可以 把 这种 积分 理论 推广 到 定义 在 完备 赋范 向量 空间 即 Banach 空间 上 的 映射 上 去 ， 这 就 为 学生 以后 学习 其它 类型 的 积分 理论 打下 一个 基础