# 共轭复数

（重定向自复共轭

## 正式定義

${\displaystyle {\overline {z}}={\overline {a+bi}}=a-bi}$

${\displaystyle z^{*}={(a+bi)}^{*}=a-bi}$

${\displaystyle {\overline {3-2i}}=3+2i}$
${\displaystyle {\overline {7}}=7}$ （實數的共軛為自身）
${\displaystyle {\overline {i}}=-i}$ （純虛數的共軛）

## 性質

${\displaystyle {\begin{array}{l}{\overline {z+w}}={\overline {z}}+{\overline {w}}\\{\overline {z-w}}={\overline {z}}-{\overline {w}}\\{\overline {zw}}={\overline {z}}\,{\overline {w}}\\{\overline {\left({\dfrac {z}{w}}\right)}}={\dfrac {\overline {z}}{\overline {w}}}&(w\neq 0)\\{\overline {z}}=z&(z\in \mathbb {R} )\\{\overline {z^{n}}}={\overline {z}}^{n}&(n\in \mathbb {Z} )\\|{\overline {z}}|=|z|\\|{\overline {z}}|^{2}=z{\overline {z}}\\{\overline {({\overline {z}})}}=z\\z^{-1}={\dfrac {\overline {z}}{|z|^{2}}}&(z\neq 0)\end{array}}}$

${\displaystyle \phi ({\overline {z}})={\overline {\phi (z)}}}$

${\displaystyle {\begin{array}{l}\exp({\overline {z}})={\overline {\exp(z)}}\\\log({\overline {z}})={\overline {\log(z)}}&(z\neq 0)\end{array}}}$

${\displaystyle {\overline {re^{i\theta }}}=re^{-i\theta }}$