Appendix/Mathematics/StepFunction: Difference between revisions
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=Unit Step Function and Its Derivative= | =Unit Step Function and Its Derivative= | ||
[ | The unit — or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] — step function, <math>H(x)</math>, is defined such that, | ||
< | <table border="0" align="center" cellpadding="8"> | ||
<tr> | |||
<td align="center"> | |||
<math> | <math> | ||
H(x) = | |||
\begin{cases} | \begin{cases} | ||
0; & ~~ | 0; & ~~ x < 0 \\ | ||
1; & ~~ | 1; & ~~ x > 0 | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
</ | </td> | ||
</tr> | |||
<tr> | |||
<td align="center"> | |||
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, §2.1 (p. 123), Eq. (2.1.6) | |||
</td> | |||
</tr> | |||
</table> | |||
As has been pointed out in, for example, [https://en.wikipedia.org/wiki/Heaviside_step_function the relevant Wikipedia discussion], the derivative of the unit step function is the Dirac Delta function, | |||
and the integral of the delta function is <math>H</math> | |||
<math> | |||
</math> | |||
<math> | where, <math>\delta(\xi)</math> is the Kronicher delta function. | ||
</math> | |||
=See Also= | =See Also= | ||
{{ SGFfooter }} | {{ SGFfooter }} | ||
Revision as of 16:16, 6 June 2022
Unit Step Function and Its Derivative
The unit — or, Heaviside — step function, , is defined such that,
|
|
|
[MF53], Part I, §2.1 (p. 123), Eq. (2.1.6) |
As has been pointed out in, for example, the relevant Wikipedia discussion, the derivative of the unit step function is the Dirac Delta function,
and the integral of the delta function is
where, is the Kronicher delta function.
See Also
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Appendices: | VisTrailsEquations | VisTrailsVariables | References | Ramblings | VisTrailsImages | myphys.lsu | ADS | |