Heaviside step distribution

The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):

$H(x)=\{\begin{array}{cc}0& x<0\\ & x=0\\ 1& x>0\end{array}$

Note that other definitions exist at $H(0)$, for example $H(0)=1$. In the famous Mathematica computer package $H(0)$ is unevaluated.

Applications

• Fourier analysis

Differentiating the Heaviside distribution

At first glance things are hopeless:

$\frac{\mathrm{d}H(x)}{\mathrm{d}x}=0,x\ne 0$

$\frac{\mathrm{d}H(x)}{\mathrm{d}x}=\mathrm{\infty },x=0$

however, lets define a less brutal jump in the form of a linear slope such that

$H_{ϵ}(x-a)=\frac{1}{ϵ}(R(x-(a-\frac{ϵ}{2}))-R(x-(a+\frac{ϵ}{2})))$

in the limit $ϵ\to 0$ this becomes the Heaviside function $H(x-a)$. However, lets differentiate first:

$\frac{\mathrm{d}}{\mathrm{d}x}{H}_{ϵ}(x-a)=\frac{1}{ϵ}(H(x-(a-\frac{ϵ}{2}))-H(x-(a+\frac{ϵ}{2})))$

in the limit this is the Dirac delta distribution. Thus

$\frac{\mathrm{d}}{\mathrm{d}x}[H(x)]=\delta (x)$.

References

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