# 符号函数

 符號函數 性質 奇偶性 奇函數 定義域 (-∞,∞) 到達域 ${\displaystyle \operatorname {sgn} x\in \{-1,0,1\}}$ 周期 N/A 特定值 當x=0 0 當x=+∞ 1 當x=-∞ -1 最大值 1 最小值 -1 其他性質 渐近线 N/A 根 0 臨界點 N/A 拐點 N/A 不動點 0,1,-1

${\displaystyle \operatorname {sgn} x=\left\{{\begin{matrix}-1&:&x<0\\0&:&x=0\\1&:&x>0\end{matrix}}\right.}$

## 性质

${\displaystyle \operatorname {sgn} x=-[x<0]+[x>0]}$

${\displaystyle x=(\operatorname {sgn} x)|x|}$

${\displaystyle \operatorname {sgn} x={x \over |x|}}$

${\displaystyle {\frac {d|x|}{dx}}={\frac {x}{|x|}}=\operatorname {sgn} x}$

${\displaystyle {d\ \operatorname {sgn} x \over dx}=2\delta (x)}$

${\displaystyle \operatorname {sgn} x=2H_{1/2}(x)-1}$

## 推广到复数

${\displaystyle \operatorname {sgn} z={\frac {z}{|z|}}}$

${\displaystyle \operatorname {sgn} z=\exp(i\arg z)\,,}$

${\displaystyle \operatorname {sgn} 0=\operatorname {sgn}(0+0i)=0.}$

${\displaystyle \operatorname {csgn} (z)={\begin{cases}1&{\text{if }}\Re (z)>0\lor (\Re (z)=0\land \Im (z)>0),\\-1&{\text{if }}\Re (z)<0\lor (\Re (z)=0\land \Im (z)<0),\\0&{\text{if }}\Re (z)=\Im (z)=0.\end{cases}}}$

${\displaystyle \operatorname {csgn} (z)={\frac {z}{\sqrt {z^{2}}}}={\frac {\sqrt {z^{2}}}{z}}.}$