Heaviside step distribution
The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):
\[ H(x) = \left\{ \begin{array}{ll} 0 & x < 0 \\ \frac{1}{2} & x=0\\ 1 & x > 0 \end{array} \right. \]
Note that other definitions exist at \(H(0)\), for example \(H(0)=1\). In the famous Mathematica computer package \(H(0)\) is unevaluated.
[edit] Applications
[edit] Differentiating the Heaviside distribution
At first glance things are hopeless:
\[\frac{{\rm d}H(x)}{{\rm d}x}= 0, ~x \neq 0\]
\[\frac{{\rm d}H(x)}{{\rm d}x}= \infty, ~x = 0\]
however, lets define a less brutal jump in the form of a linear slope such that
\[H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( R(x - (a-\frac{\epsilon}{2})) - R (x - (a+\frac{\epsilon}{2}))\right)\]
in the limit \(\epsilon \rightarrow 0\) this becomes the Heaviside function \(H(x-a)\). However, lets differentiate first:
\[\frac[[:Template:\rm d]]{{\rm d}x} H_{\epsilon}(x-a)= \frac{1}{\epsilon}\left( H(x - (a-\frac{\epsilon}{2})) - H (x - (a+\frac{\epsilon}{2}))\right)\]
in the limit this is the Dirac delta distribution. Thus
\[\frac[[:Template:\rm d]]{{\rm d}x} [H(x)]= \delta(x)\].
