Appendix/Mathematics/StepFunction - Revision history

Jet53man: /* Unit Step Function and Its Derivative */

2023-08-12T01:12:38Z

Unit Step Function and Its Derivative

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 =Unit Step Function and Its Derivative=
 The unit &#8212; or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] &#8212; step function, <math>H(x)</math>, is defined such that,
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 <math>\int_{-\infty}^x \delta(\xi)d\xi    \, .</math>
    </td>
 </tr>
 </table>

Jet53man: /* Unit Step Function and Its Derivative */

2022-06-07T20:52:56Z

Unit Step Function and Its Derivative

← Older revision		Revision as of 16:52, 7 June 2022
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	\end{cases}		\end{cases}
	</math>		</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>		</td>
	</tr>		</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>		</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,		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,

Jet53man: /* Unit Step Function and Its Derivative */

2022-06-07T18:31:51Z

Unit Step Function and Its Derivative

← Older revision		Revision as of 14:31, 7 June 2022
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	</tr>		</tr>
	</table>		</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~~,		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>

	~~and the integral of the delta function is~~ <math>H</math>		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">

	~~where,~~ <math>\delta(\xi)</math> ~~is the Kronicher delta function.~~		<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>

	=See Also=		=See Also=

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Jet53man at 20:16, 6 June 2022

2022-06-06T20:16:06Z

← Older revision		Revision as of 16:16, 6 June 2022
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	=Unit Step Function and Its Derivative=		=Unit Step Function and Its Derivative=

	[~~<b>[[Appendix~~/~~References~~#~~MF53\|~~<~~font color="red"~~>~~MF53~~</~~font~~>~~]]</b>], Part I~~, ~~§2.1 (p. 123)~~, ~~Eq. (2.1.6)~~		The unit — or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] — step function, <math>H(x)</math>, is defined such that,

	<~~div~~ align="center">		<table border="0" align="center" cellpadding="8">
			<tr>
			<td align="center">
	<math>		<math>
	u(z) = ~~\int_{-\infty}^\infty \delta(\xi)d\xi =~~		H(x) =
	\begin{cases}		\begin{cases}
	0; & ~~z < 0 \\		0; & ~~ x < 0 \\
	1; & ~~ z > 0		1; & ~~ x > 0
	\end{cases}		\end{cases}
	</math>		</math>
	</~~div~~>		</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,

	~~<!-- WORKS~~
	~~<math>~~
	~~f(n) =~~
	~~\begin{cases}~~
	~~n/2, & \text{if }n\text{ is even} \\~~
	~~3n+1, & \text{if }n\text{ is old}~~
	~~\end{cases}~~
	~~</math>~~
	~~-->~~

	~~<!-- EARLY TRIES THAT DID NOT WORK ...~~		and the integral of the delta function is <math>H</math>
	~~<math>~~
	~~\begin{equation}~~
	~~f(x)=\begin{cases}~~
	~~1, & \text{if $x<0$}.\\~~
	~~0, & \text{otherwise}.~~
	~~\end{cases}~~
	~~\end{equation}~~
	~~</math>~~

	<math>
	~~\left\{~~
	~~\begin{array}{1}~~
	~~0\\~~
	1
	~~\end{array}~~
	~~\right.~~
	</math>

	<math>		where, <math>\delta(\xi)</math> is the Kronicher delta function.
	f(x)~~=\left\{~~
	~~\begin{array}{ll}~~
	~~1, & \mbox{if $x<0$}.\\~~
	~~0, & \mbox{otherwise}.~~
	~~\end{array}~~
	~~\right.~~
	</math>
	~~-->~~

	=See Also=		=See Also=

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Jet53man: /* Step Function and its Derivative */

2022-06-05T23:07:04Z

Step Function and its Derivative

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+[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, &sect;2.1 (p. 123), Eq. (2.1.6)
 =See Also=
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Jet53man: Created page with "FORCETOC =Step Function and its Derivative= To be written … =See Also= {{ SGFfooter }}"

2022-06-05T22:38:58Z

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=Step Function and its Derivative=

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Appendix/Mathematics/StepFunction - Revision history

Jet53man: /* Unit Step Function and Its Derivative */

Jet53man: /* Unit Step Function and Its Derivative */

Jet53man: /* Unit Step Function and Its Derivative */

Jet53man at 20:16, 6 June 2022

Jet53man: /* Step Function and its Derivative */

Jet53man: Created page with "__FORCETOC__ =Step Function and its Derivative= To be written … =See Also= {{ SGFfooter }}"

Jet53man: Created page with "FORCETOC =Step Function and its Derivative= To be written … =See Also= {{ SGFfooter }}"