Heaviside step distribution

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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[edit]

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



  1. Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.