# 单位阶跃函数

${\displaystyle H[n]={\begin{cases}0,&n<0,\\1,&n\geq 0,\end{cases}}}$

${\displaystyle H(x)={\begin{cases}0,&x<0\\{\frac {1}{2}},&x=0\\1,&x>0\end{cases}}}$

${\displaystyle H(x)={\frac {1}{2}}\left(1+\operatorname {sgn}(x)\right)}$

## 連續函數逼近

• ${\displaystyle H(x)=\lim _{k\rightarrow \infty }{\frac {1}{2}}(1+\tanh kx)=\lim _{k\rightarrow \infty }{\frac {1}{1+\mathrm {e} ^{-2kx}}}}$

• ${\displaystyle H(x)=\lim _{k\rightarrow \infty }{\frac {1}{2}}+{\frac {1}{\pi }}\arctan(kx)\ }$
• ${\displaystyle H(x)=\lim _{k\rightarrow \infty }{\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} (kx)\ }$

## 積分表示

{\displaystyle {\begin{aligned}H(x)&=\lim _{\varepsilon \to 0^{+}}-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau +i\varepsilon }}e^{-ix\tau }d\tau \\&=\lim _{\varepsilon \to 0^{+}}{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {1}{\tau -i\varepsilon }}e^{ix\tau }d\tau .\end{aligned}}}

## 參考資料

1. ^ Weisstein, Eric W. (编). Heaviside Step Function. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.