|
|
| Line 4: |
Line 4: |
| H(x) = \left\{ | | H(x) = \left\{ |
| \begin{array}{ll} | | \begin{array}{ll} |
| 0 & x < 0 \\ | | 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
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| | |
| :<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
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| <math>H(x-a)</math>. However, lets differentiate first:
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| | |
| :<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]]
| |
Revision as of 02:28, 5 July 2007
The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):
- <math>
H(x) = \left\{
\begin{array}{ll}
0