Appendix/Mathematics/StepFunction - Revision history

Joel2: Created page with "FORCETOC =Unit Step Function and Its Derivative= ==Standard Presentation== The unit — or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] — step function, $H(x)$ , is defined such that,

$H(x) = \begin{cases} 0; & x < 0 \\ 1; & x > 0 \end{cases}$

[Appendix/References#MF53|
2024-07-08T18:38:16Z

Created page with "FORCETOC  =Unit Step Function and Its Derivative= ==Standard Presentation== 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> H(x) = \begin{cases} 0; & x < 0 \\ 1; & x > 0 \end{cases} </math> <p><br /> [<b>Appendix/References#MF53|<fon..."

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FORCETOC


=Unit Step Function and Its Derivative=

==Standard Presentation==
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>
H(x) =
\begin{cases}
0; & x < 0 \\
1; & x > 0
\end{cases}
</math>
<p><br />
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, §2.1 (p. 123), Eq. (2.1.6)
</td>
<td align="center" rowspan="2">
[[File:Heaviside01.png|300px|Heaviside Function]]
</td>
</tr>
</table>
In evaluating this function at <math>x=0</math>, we will adopt the ''half-maximum convention'' and set <math>H(0) = \tfrac{1}{2}</math>. As has been pointed out in, for example, [https://en.wikipedia.org/wiki/Heaviside_step_function a relevant Wikipedia discussion], the derivative of the unit step function is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{dH(x)}{dx}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\delta(x) \, ,</math>
</td>
</tr>
</table>

where, <math>\delta(x)</math> is the Dirac Delta function. Hence, the unit step function is sometimes written as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>H(x)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\int_{-\infty}^x \delta(\xi)d\xi \, .</math>
</td>
</tr>
</table>

==Sign of a Function==

Notice that the sign of <math>x</math>, may be written in terms of the step function as,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\sgn[x] = \frac{|x|}{x}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2H(x) - 1 \, .</math>
</td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{d}{dx} \biggl[ \sgn(x)\biggr] = \frac{d}{dx}\biggl[ \frac{|x|}{x} \biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>2\delta(x) \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">{{ Hunter2003 }}, §2.2, immediately following Eq. (3)</td>
</tr>
</table>

=See Also=

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