Heaviside step distribution
The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):
Note that other definitions exist at , for example . In the famous Mathematica computer package is unevaluated.
Differentiating the Heaviside distribution
At first glance things are hopeless:
however, lets define a less brutal jump in the form of a linear slope such that
in the limit this becomes the Heaviside function . However, lets differentiate first:
in the limit this is the Dirac delta distribution. Thus