Examples

Next: The operators and Up: 2: THE STANDARD DISCONTINUITIES Previous: The simplest properties

Examples

	$\begin{eqnarraystar} e^{-\alpha t}H(t) & \stackrel{0}{\sim}& H(t)\equiv R_0(t) \... ...el{2}{\sim}&R_0(t)-\alpha R_1(t)+\alpha^{2}R_2(t)\\ & \cdots&\end{eqnarraystar}$

$\begin{eqnarraystar} e^{-\alpha t}\sin(\omega_0t)H(t) & \stackrel{0}{\sim}&0\\ ... ...}{\sim}&\omega_{0}R_1(t)+2\alpha\omega_{0}R_2(t)\\ & \cdots &\end{eqnarraystar}$

$\begin{eqnarraystar} te^{-\alpha t}\cos(\omega_0t)H(t) & \stackrel{0}{\sim}&0\\ ... ...t)\\ & \stackrel{2}{\sim}&R_1(t)-2\alpha R_2(t)\\ & \cdots &\end{eqnarraystar}$

From equation (10), it follows immediately that:

$\begin{displaymath} {\bf D}^pS(t)R_q(t) \stackrel{q-p}{\sim} S(t)R_{q-p}(t) .\end{displaymath}$

Indeed:

	$\begin{eqnarraystar} {\bf D}^pS(t)R_q(t) & \stackrel{q-p}{\sim} {\bf D}^pS(0)R_{... ... & \cdots =S(0)R_{q-p}(t) \stackrel{q-p}{\sim}S(t)R_{q-p}(t).\end{eqnarraystar}$

Next: The operators and Up: 2: THE STANDARD DISCONTINUITIES Previous: The simplest properties

Stanford Exploration Project
1/13/1998