Revision as of 07:55, 23 December 2008 by imported>Paul Wormer
In mathematics, physics, and engineering the Heaviside step function is the following
function,
![{\displaystyle H(x)={\begin{cases}1&\quad {\hbox{if}}\quad x>0\\{\frac {1}{2}}&\quad {\hbox{if}}\quad x=0\\0&\quad {\hbox{if}}\quad x<0\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18df9f83b9069b7539aef0192ead37d4ce47492d)
Note that a block function BΔ of width Δ and height 1/Δ can be given in terms of step functions (for positive Δ), namely
![{\displaystyle B_{\Delta }(x)={\begin{cases}{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {0-0}{\Delta }}=0&\quad {\hbox{if}}\quad x<-\Delta /2\\{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {1-0}{\Delta }}={\frac {1}{\Delta }}&\quad {\hbox{if}}\quad -\Delta /2<x<\Delta /2\\{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {1-1}{\Delta }}=0&\quad {\hbox{if}}\quad x>\Delta /2\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc32cca4f7fabfe1408982ced7e30629240c7f56)
The derivative of the step function is
![{\displaystyle H'(x)=\lim _{\Delta \rightarrow 0}{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}=\lim _{\Delta \rightarrow 0}B_{\Delta }(x)=\delta (x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9cd64adf15a2b86933fc9634ad09af2777cee0)
where δ(x) is the Dirac delta function, which may be defined as the block function in the limit of zero width, see this article.