Heaviside step distribution - Revision history

Carl McBride: Reverted edits by 89.20.145.223 (Talk); changed back to last version by Carl McBride

2007-07-05T11:12:49Z

Reverted edits by 89.20.145.223 (Talk); changed back to last version by Carl McBride

@@ Line 4: / Line 4: @@
 H(x) = \left\{
 \begin{array}{ll}
+          &  x < 0 \\

89.20.145.223: Replacing page with 'The '''Heaviside step distribution''' is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020): : $H(x) = \left\{
\begin{array}{ll}
0'$

2007-07-05T01:28:25Z Replacing page with 'The '''Heaviside step distribution''' is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020): :<math> H(x) = \left\{ \begin{array}{ll} 0'

@@ Line 4: / Line 4: @@
 H(x) = \left\{
 \begin{array}{ll}
-          &  x < 0 \\

Carl McBride at 09:32, 30 May 2007

2007-05-30T09:32:20Z

← Older revision		Revision as of 11:32, 30 May 2007
Line 9:		Line 9:
	\end{array} \right.		\end{array} \right.
	</math>		</math>

			Note that other definitions exist at <math>H(0)</math>, for example <math>H(0)=1</math>.
			In the famous [http://www.wolfram.com/products/mathematica/index.html Mathematica] computer
			package <math>H(0)</math> is unevaluated.

			==Applications==
			*[[Fourier analysis]]
	==Differentiating the Heaviside distribution==		==Differentiating the Heaviside distribution==
	At first glance things are hopeless:		At first glance things are hopeless:

Carl McBride at 09:34, 29 May 2007

2007-05-29T09:34:52Z

← Older revision		Revision as of 11:34, 29 May 2007
Line 26:		Line 26:
	:<math>\frac{{\rm d}}{{\rm d}x} H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( H(x - (a-\frac{\epsilon}{2})) - H (x - (a+\frac{\epsilon}{2}))\right)</math>		:<math>\frac{{\rm d}}{{\rm d}x} H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( H(x - (a-\frac{\epsilon}{2})) - H (x - (a+\frac{\epsilon}{2}))\right)</math>

	in the limit this is the [[Dirac delta ~~function~~]]. Thus		in the limit this is the [[Dirac delta distribution]]. Thus

	~~:<math>\frac{{\rm d}}{{\rm d}x} [H(x)]= \delta(x)</math>~~

	~~The delta function has the fundamental property that~~

	~~:<math>\int_{-\infty}^{\infty} f(x) \delta(x-a) {\rm d}x = f(a)</math>~~


			:<math>\frac{{\rm d}}{{\rm d}x} [H(x)]= \delta(x)</math>.
	==References==		==References==
	#[http://store.doverpublications.com/0486612724.html Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.]		#[http://store.doverpublications.com/0486612724.html Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.]
	[[category:mathematics]]		[[category:mathematics]]

Carl McBride at 09:32, 29 May 2007

2007-05-29T09:32:19Z

@@ Line 9: / Line 9: @@
 \end{array} \right.
 </math>
 ==References==
 #[http://store.doverpublications.com/0486612724.html  Milton Abramowitz and  Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.]
 [[category:mathematics]]

Carl McBride at 09:38, 28 May 2007

2007-05-28T09:38:32Z

← Older revision		Revision as of 11:38, 28 May 2007
Line 1:		Line 1:
	The '''Heaviside step distribution''' is defined by		The '''Heaviside step distribution''' is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):

	:<math>		:<math>
Line 9:		Line 9:
	\end{array} \right.		\end{array} \right.
	</math>		</math>
			==References==
			#[http://store.doverpublications.com/0486612724.html Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.]
	[[category:mathematics]]		[[category:mathematics]]

Carl McBride: Reverted edits by Cuesta (Talk); changed back to last version by Nice and Tidy

2007-05-28T09:32:25Z Reverted edits by Cuesta (Talk); changed back to last version by Nice and Tidy

← Older revision		Revision as of 11:32, 28 May 2007
Line 5:		Line 5:
	\begin{array}{ll}		\begin{array}{ll}
	0 & x < 0 \\		0 & x < 0 \\
			\frac{1}{2} & x=0\\
	1 & x > 0		1 & x > 0
	\end{array} \right.		\end{array} \right.
	</math>		</math>
	[[category:mathematics]]		[[category:mathematics]]

Cuesta at 15:10, 26 May 2007

2007-05-26T15:10:29Z

← Older revision		Revision as of 17:10, 26 May 2007
Line 5:		Line 5:
	\begin{array}{ll}		\begin{array}{ll}
	0 & x < 0 \\		0 & x < 0 \\
	~~\frac{1}{2} & x=0\\~~
	1 & x > 0		1 & x > 0
	\end{array} \right.		\end{array} \right.
	</math>		</math>
	[[category:mathematics]]		[[category:mathematics]]

Nice and Tidy at 16:06, 25 May 2007

2007-05-25T16:06:46Z

← Older revision		Revision as of 18:06, 25 May 2007
Line 9:		Line 9:
	\end{array} \right.		\end{array} \right.
	</math>		</math>
			[[category:mathematics]]

Carl McBride: New page: The '''Heaviside step distribution''' is defined by : $H(x) = \left\{ \begin{array}{ll} 0 & x < 0 \\ \frac{1}{2} & x=0\\ 1 & x > 0 \end{array} \right.$

2007-03-21T14:37:45Z

New page: The '''Heaviside step distribution''' is defined by :<math> H(x) = \left\{ \begin{array}{ll} 0 & x < 0 \\ \frac{1}{2} & x=0\\ 1 & x > 0 \end{array} \right. </math>

New page

The '''Heaviside step distribution''' is defined by

:<math>
H(x) = \left\{
\begin{array}{ll}
0 & x < 0 \\
\frac{1}{2} & x=0\\
1 & x > 0
\end{array} \right.
</math>