# 复分析小品-生成函数

In mathematics, a generating function is a way of encoding an infinite sequence of numbers ($a_n$) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence.

Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the “variable” remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

# 复分析小品-分式线性变换

$$f(z) = \frac{az+b}{cz+d}$$

• 分式线性变换的复合还是一个分式线性变换变换.
• 任何分式线性变换都可以表示为三种简单变换的复合, 即:平移、旋转、反演的复合.

# 线性代数-矩阵基础-3

## 矩阵的秩

**【定理】** 初等变换不改变矩阵的秩.进一步的,初等行(列)变换不改变矩阵任意列(行)之间的线性相关性.

# 线性代数-矩阵基础 2

## 矩阵的初等变换与分块运算

**定义**【初等变换】 以下三种对矩阵的操作, 每一种都称为对矩阵的进行的一次 **初等变换**. - 将矩阵的一行(列)与另一行(列)交换位置; - 将矩阵的某一行乘以一个非零倍数; - 将矩阵的一行(列)加到另一行(列)上去.

[定义][导数]

# 微积分之数列与级数

## 序列与极限

$$\lim_{n\rightarrow\infty}x_{n} = x.$$

# Review - $\mathbb{R}^n$ 上的拓扑(1) 这篇笔记主要内容是回顾 $\mathbb{R}^n$ 上的拓扑. 事实上,这里是要对由度量诱导的 $\mathbb{R}^n$ 的度量空间上的拓扑进行总结. 对于一般的拓扑空间(不依赖特定度量的性质)的拓扑性质, 更为详细的内容, 可以参考任意一本《点集拓扑学》或《一般拓扑学》之类的讲义. 事实上, 度量空间上的极限, 连续性, 和紧性都是空间拓扑性质的例子.

# Review-微积分之【函数定义】

Mathematics links the abstract world of mental concepts to the real world of physical things without being located in completely in either.
**Ian Stewart** — Preface to second edition of *What is Mathematics?* by **Richard Courant** and **Herbert Robbins**, revised by **Ian Stewart** (1996).