Editing Heaviside step distribution
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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> | :<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 | in the limit this is the [[Dirac delta function]]. 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> | |||
==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]] |