Difference between revisions of "Derivation of delta exponential T=isolated points"

From timescalewiki
Jump to: navigation, search
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
For $t>s$, find [[Delta exponential|$e_p$]] by computing
+
Let $\mathbb{T}$ be a [[time scale]] of [[isolated points]]. For $t>s$, find [[Delta exponential|$e_p$]] by computing
 
$$\begin{array}{ll}
 
$$\begin{array}{ll}
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1+\mu(\tau)p(\tau)) \Delta \tau \right) \\
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1+\mu(\tau)p(\tau)) \Delta \tau \right) \\
&= \exp \left( \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \log(1+\mu(t_k)p(t_k) \right) \\
+
&= \exp \left( \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \log(1+\mu(t_k)p(t_k)) \right) \\
 
&= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} 1+\mu(t_k)p(t_k).
 
&= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} 1+\mu(t_k)p(t_k).
 
\end{array}$$
 
\end{array}$$

Latest revision as of 23:35, 9 June 2015

Let $\mathbb{T}$ be a time scale of isolated points. For $t>s$, find $e_p$ by computing $$\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1+\mu(\tau)p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\pi(s)}^{\pi(t)-1} \log(1+\mu(t_k)p(t_k)) \right) \\ &= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} 1+\mu(t_k)p(t_k). \end{array}$$