The simplest properties

Let $S(t) \in {\rm C}^{q}$, then:

$$R_{q}(t) + S(t) \stackrel{q}{\sim} R_{q}(t)$$

and  

$$S(t)R_{q}(t) \stackrel{q}{\sim} S(0) R_{q}(t).$$ (10)

Let us prove the last formula. We must prove that ${\bf D}^{q}[S(t)R_{q}(t) -S(0)R_{q}(t)]$ is a continuous function:

$$\begin{eqnarray} {\bf D}^{q}[\cdot ]& = & {\bf D}^{q} \{[S(t) - S(0)] R_{q}(t) \... ...{S^{'}(t) R_{q}(t) \} + {\bf D}^{q-1}\{ [S(t)-S (0)]R_{q-1}(t) \}.\end{eqnarray}$$ (11) (12) (13)

The last term has the sharpest discontinuity , so we needn't pay any attention to the first one. Repeating the procedure we get at the end the sharpest term:

[S(t)-S(0)]R₀(t)

which is continuous at the point t=0.

Instead of equation (10), we may derive a more general formula:

$$S(t)R_q(t)\left\{\begin{array} {cl} \stackrel{q}{\sim} & S(... ...}^{s}{(q+r)!\over r!q!}S^{(r)}(0)R_{q+r}(t).\end{array} \right.$$