Heaviside step distribution: Difference between revisions
(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') |
Carl McBride (talk | contribs) m (Reverted edits by 89.20.145.223 (Talk); changed back to last version by Carl McBride) |
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H(x) = \left\{ | H(x) = \left\{ | ||
\begin{array}{ll} | \begin{array}{ll} | ||
0 | 0 & x < 0 \\ | ||
\frac{1}{2} & x=0\\ | |||
1 & x > 0 | |||
\end{array} \right. | |||
</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== | |||
At first glance things are hopeless: | |||
:<math>\frac{{\rm d}H(x)}{{\rm d}x}= 0, ~x \neq 0</math> | |||
:<math>\frac{{\rm d}H(x)}{{\rm d}x}= \infty, ~x = 0</math> | |||
however, lets define a less brutal jump in the form of a linear slope | |||
such that | |||
:<math>H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( R(x - (a-\frac{\epsilon}{2})) - R (x - (a+\frac{\epsilon}{2}))\right)</math> | |||
in the limit <math>\epsilon \rightarrow 0</math> this becomes the Heaviside function | |||
<math>H(x-a)</math>. However, lets differentiate first: | |||
:<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 distribution]]. Thus | |||
:<math>\frac{{\rm d}}{{\rm d}x} [H(x)]= \delta(x)</math>. | |||
==References== | |||
#[http://store.doverpublications.com/0486612724.html Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.] | |||
[[category:mathematics]] | |||
Latest revision as of 12:12, 5 July 2007
The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(x) = \left\{ \begin{array}{ll} 0 & x < 0 \\ \frac{1}{2} & x=0\\ 1 & x > 0 \end{array} \right. }
Note that other definitions exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(0)} , for example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(0)=1} . In the famous Mathematica computer package Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(0)} is unevaluated.
Applications[edit]
Differentiating the Heaviside distribution[edit]
At first glance things are hopeless:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\rm d}H(x)}{{\rm d}x}= 0, ~x \neq 0}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\rm d}H(x)}{{\rm d}x}= \infty, ~x = 0}
however, lets define a less brutal jump in the form of a linear slope such that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( R(x - (a-\frac{\epsilon}{2})) - R (x - (a+\frac{\epsilon}{2}))\right)}
in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon \rightarrow 0} this becomes the Heaviside function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(x-a)} . However, lets differentiate first:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)}
in the limit this is the Dirac delta distribution. Thus
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\rm d}}{{\rm d}x} [H(x)]= \delta(x)} .