Difference between revisions of "Derivation of nabla exponential T=isolated points"
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(Created page with "Let $\mathbb{T}$ be a time scale of isolated points. For $t>s$, find $\hat{e}_p$ by computing $$\begin{array}{ll} \hat{e}_p(t,s) &= \exp \left(-\...") |
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\hat{e}_p(t,s) &= \exp \left(-\displaystyle\int_s^t \dfrac{1}{\nu(\tau)} \log(1-\nu(\tau)p(\tau)) \Delta \tau \right) \\ | \hat{e}_p(t,s) &= \exp \left(-\displaystyle\int_s^t \dfrac{1}{\nu(\tau)} \log(1-\nu(\tau)p(\tau)) \Delta \tau \right) \\ | ||
&= \exp \left(-\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \log(1-\nu(t_k)p(t_k)) \right) \\ | &= \exp \left(-\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \log(1-\nu(t_k)p(t_k)) \right) \\ | ||
| − | &= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} \dfrac{1}{1 | + | &= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} \dfrac{1}{1-\nu(t_k)p(t_k)}. |
\end{array}$$ | \end{array}$$ | ||
Latest revision as of 23:39, 9 June 2015
Let $\mathbb{T}$ be a time scale of isolated points. For $t>s$, find $\hat{e}_p$ by computing $$\begin{array}{ll} \hat{e}_p(t,s) &= \exp \left(-\displaystyle\int_s^t \dfrac{1}{\nu(\tau)} \log(1-\nu(\tau)p(\tau)) \Delta \tau \right) \\ &= \exp \left(-\displaystyle\sum_{k=\pi(s)+1}^{\pi(t)} \log(1-\nu(t_k)p(t_k)) \right) \\ &= \displaystyle\prod_{k=\pi(s)}^{\pi(t)-1} \dfrac{1}{1-\nu(t_k)p(t_k)}. \end{array}$$
