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

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For $t>s$, find [[Delta exponential|$e_p$]] by computing
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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) \\

Revision as of 23:27, 9 June 2015

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