Heaviside function

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1: 1.16 Distributions
§1.16(iv) HeavisideFunction
1.16.13 $H\left(x\right)=\begin{cases}1,&x>0,\\ 0,&x\leq 0.\end{cases}$
1.16.14 $H\left(x-x_{0}\right)=\begin{cases}1,&x>x_{0},\\ 0,&x\leq x_{0}.\end{cases}$
1.16.44 $\operatorname{sign}\left(x\right)=2H\left(x\right)-1,$ $x\neq 0$,
where $H\left(x\right)$ is the Heaviside function defined in (1.16.13), and the derivatives are to be understood in the sense of distributions. …
2: 18.40 Methods of Computation
$H\left(x\right)$ being the Heaviside step-function, see (1.16.13). …
3: 2.6 Distributional Methods
2.6.37 $F\ast G=D^{n+m}(f\ast g).$
Furthermore, $K^{+}$ contains the distributions $H$, $\delta$, and $t^{\lambda}$, $t>0$, for any real (or complex) number $\lambda$, where $H$ is the distribution associated with the Heaviside function $H\left(t\right)$1.16(iv)), and $t^{\lambda}$ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of $\lambda$. …
4: 1.14 Integral Transforms
1.14.22 $\mathscr{L}\mskip-3.0muf_{a}^{+}\mskip 3.0mu\left(s\right)={\mathrm{e}}^{-as}% \mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right),$
where $f^{+}_{a}(t)=H\left(t-a\right)f(t-a)$ and $H$ is the Heaviside function; see (1.16.13). …
5: 1.4 Calculus of One Variable
If, for example, $\alpha(x)=H\left(x-x_{n}\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $\,\mathrm{d}\alpha(x)$ is $\delta\left(x-x_{n}\right)\,\mathrm{d}x$, where $\delta\left(x-x_{n}\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=f(x_{n})$ for $x_{n}\in(a,b)$ and $0$ otherwise. …