Definition

We shall call a standard discontinuity a function:

$$R_{q}(t)= {t_{+}^{q}\over q!}, \mbox{\hspace{1.0cm}} (q=0,1,2,\ldots;\; t_+=tH(t)).$$

We see that $H(t) \equiv R_{0}(t)$ and $t_{+} \equiv R_{1}(t)$.

The denominator q! is introduced in order to have a very simple (and very important!) relation:

$${\bf D}R_{q}(t)=R_{q-1}(t), \mbox{\hspace{1.0cm}} q = 1,2,3,\ldots$$

where ${\bf D} = {d\over dt}$.

Since ${\bf D}R_{0}(t)= \delta(t)$, it is convenient to define

$$R_{-1}(t) = \delta(t)$$

(and $R_{-k}(t) = \delta^{(k)}(t), \ \ \ k= 1,2,\ldots$).