$$
\global\long\def\cc{\mathbb{C}}%
\global\long\def\rr{\mathbb{R}}%
\global\long\def\bbq{\mathbb{Q}}%
\global\long\def\bbz{\mathbb{Z}}%
\global\long\def\bbn{\mathbb{N}}%
\global\long\def\bbd{\mathbb{D}}%
\global\long\def\cco{\mathcal{O}}%
\global\long\def\Del{\varDelta}%
\global\long\def\del{\delta}%
\global\long\def\Ome{\Omega}%
\global\long\def\lam{\lambda}%
\global\long\def\eps{\varepsilon}%
\global\long\def\vphi{\varphi}%
\global\long\def\ii{\mathrm{i}\,}%
\global\long\def\oo#1{\mathrm{o}\left(#1\right)}%
\global\long\def\half{\frac{1}{2}}%
\global\long\def\ra{\rightarrow}%
\global\long\def\Ra{\longrightarrow}%
\global\long\def\era{\mapsto}%
\global\long\def\Era{\longmapsto}%
\global\long\def\deff{\coloneqq}%
\global\long\def\fed{\eqqcolon}%
\global\long\def\sub{\subset}%
\global\long\def\con{\supset}%
\global\long\def\re{\mathrm{Re}}%
\global\long\def\im{\mathrm{Im}}%
\global\long\def\dis#1{D\left(#1\right)}%
\global\long\def\bb#1{\left(#1\right)}%
\global\long\def\abs#1{\left|#1\right|}%
\global\long\def\Bar#1{\overline{#1}}%
\global\long\def\pp#1#2{\frac{\partial#1}{\partial#2}}%
\global\long\def\ppb#1#2{\frac{\partial#1}{\partial\bar{#2}}}%
\global\long\def\dd#1#2{\frac{\mathrm{d}#1}{\mathrm{d}#2}}%
$$
复可微和Cauchy–Riemann条件
（1） 我们称\(f:\dis{z_{0},r}\ra\cc\)在\(z_{0}\)处复可微（或复可导），若以下极限存在： \[
f^{\prime}\bb{z_{0}}\deff\lim_{\del z\ra0}\frac{f\bb{z_{0}+\Del z}-f\bb{z_{0}}}{\Del z}.
\] \(f^{\prime}\bb{z_{0}}\) 称为 \(f\) 在 \(z_{0}\) 处的导数。
（2） 设\(\Ome\sub\cc\)为开集，若函数\(f:\Ome\ra\cc\) 在\(\Ome\)中每一点处皆为复可微，则称\(f\)在\(\Ome\)上全纯（也称解析）。\(\Ome\) 上所有全纯函数构成的集合记为 \(\cco\left(\Ome\right)\)。称\(f\)在一点 \(a\) 处全纯，如果 \(f\) 在\(a\)的某个邻域上全纯。这样的函数全体记为 \(\cco_{a}\)。