Questions in category: 复分析 (Complex Analysis)

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## 1. [Def] 施瓦茨导数(Schwarzian derivative)

Posted by haifeng on 2024-04-21 10:05:40 last update 2024-05-06 14:27:57 | Answers (2) | 收藏

$\{f,z\}:=\frac{f'''(z)}{f'(z)}-\frac{3}{2}\biggl(\frac{f''(z)}{f'(z)}\biggr)^2$

$\biggl(\frac{f''(z)}{f'(z)}\biggr)'=\frac{f'''(z)f'(z)-f''(z)f''(z)}{(f'(z))^2}=\frac{f'''(z)}{f'(z)}-\biggl(\frac{f''(z)}{f'(z)}\biggr)^2,$

$\{f,z\}=\biggl(\frac{f''(z)}{f'(z)}\biggr)'-\frac{1}{2}\biggl(\frac{f''(z)}{f'(z)}\biggr)^2.$

$g(z)=\frac{af(z)+b}{cf(z)+d},$

$\frac{g'''(z)}{g'(z)}-\frac{3}{2}\biggl(\frac{g''(z)}{g'(z)}\biggr)^2=\frac{f'''(z)}{f'(z)}-\frac{3}{2}\biggl(\frac{f''(z)}{f'(z)}\biggr)^2.$

$\{f,t\}=\varphi^{-2}\cdot\Bigl(\{f,z\}+2A^2-2A'\Bigr)$

References:

## 2. 对于复变量 $z\in\mathbb{C}$, 函数 $\sin z$ 有界吗?

Posted by haifeng on 2023-06-29 08:32:46 last update 2023-06-29 09:13:00 | Answers (0) | 收藏

$e^z=1+\frac{z^1}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots+\frac{z^n}{n!}+\cdots$

$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\cdots$

$\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\cdots$

$e^{iz}=\cos z+i\sin z\tag{1}$

$e^{-iz}=\cos z-i\sin z.\tag{2}$

(1) 和 (2) 相乘, 得

$1=e^{iz}\cdot e^{-iz}=\cos^2 z+\sin^2 z.$

## 3. 复分析概要

Posted by haifeng on 2022-11-05 08:59:53 last update 2022-11-05 08:59:53 | Answers (0) | 收藏

Cauchy 积分公式  ==>  全纯函数的均值性质  ==> 最大模原理  ==> Schwarz 引理

Cauchy 积分公式

$f(z)=\frac{1}{2\pi i}\int_{\partial D(z_0, r)}\frac{f(\zeta)}{\zeta-z_0}\mathrm{d}\zeta$

$f(z)=\frac{1}{2\pi}\int_{0}^{2\pi}f(z_0+re^{i\theta})\mathrm{d}\theta$

## 4. 证明下面的复函数是连续的.

Posted by haifeng on 2021-07-03 09:14:40 last update 2021-07-03 09:16:19 | Answers (0) | 收藏

$\varphi(z)=\begin{cases} z, & |z|\leqslant R,\\ \frac{Rz}{|z|}, & |z| > R, \end{cases}$

## 5. 求下面方程的解

Posted by haifeng on 2021-03-22 11:20:17 last update 2021-03-22 11:35:42 | Answers (0) | 收藏

$f(s)=\int_{0}^{T}(e^{i\omega t-its}-e^{i\omega t+its})\mathrm{d}t,$

Remark:

## 6. [Def] 椭圆函数

Posted by haifeng on 2020-12-31 21:15:29 last update 2020-12-31 21:16:48 | Answers (0) | 收藏

References:

## 7. 复变函数 $\mathrm{Ln}z$ 或 $\mathrm{Log}z$ 的定义.

Posted by haifeng on 2020-12-06 09:32:47 last update 2020-12-06 09:56:46 | Answers (1) | 收藏

$e^{\omega+i2k\pi}=e^{\omega}.$

$z=e^{\omega}=e^{u+iv}=e^u\cdot e^{iv}=e^u(\cos v+i\sin v)$

$f(z)=\mathrm{Ln}z=u+iv=\ln r+i(\theta+2k\pi)$

$\frac{\mathrm{d}}{\mathrm{d}z}f(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}.$

Claim.

$\frac{\mathrm{d}}{\mathrm{d}z}\mathrm{Ln}z=\frac{1}{z}$

References:

[1]  A. D. 亚历山大洛夫  等 著 《数学，它的内容、方法和意义》,  科学出版社

## 8. 写出 $\frac{e^{it}+z}{e^{it}-z}$ 的实部和虚部

Posted by haifeng on 2019-09-05 12:49:38 last update 2019-09-05 13:13:42 | Answers (1) | 收藏

$\mathrm{Re}(\frac{e^{it}+z}{e^{it}-z})=\frac{1-r^2}{1-2r\cos(\theta-t)+r^2},$

$\mathrm{Im}(\frac{e^{it}+z}{e^{it}-z})=\frac{2r\sin(\theta-t)}{1-2r\cos(\theta-t)+r^2}.$

References:

W. Rudin, 《实分析和复分析》

## 9. 若 $f(z)=\sum_{n=0}^{\infty}c_n z^n$ 在收敛圆周上只有一个奇点 $z_0$, 且这个奇点是一级极点, 则 $\lim\limits_{n\rightarrow\infty}\frac{c_n}{c_{n+1}}=z_0$.

Posted by haifeng on 2016-10-07 08:57:55 last update 2016-10-07 08:57:55 | Answers (1) | 收藏

$\lim\limits_{n\rightarrow\infty}\frac{a_n}{a_{n+1}}=z_0.$

## 10. Hartogs 扩张定理

Posted by haifeng on 2015-04-24 09:48:43 last update 2015-04-24 09:48:43 | Answers (0) | 收藏

Friedrich Hartogs 在 1906 年发现了这样一个事实.

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