Difference between revisions of "Integers"
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(Created page with "The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a time scale. {| class="wikitable" |+$\mathbb{T}=h\mathbb{Z}$ |- |Generic element $t...") |
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| − | The set $ | + | The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a [[time scale]]. |
{| class="wikitable" | {| class="wikitable" | ||
Revision as of 19:45, 19 May 2014
The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.
| Generic element $t\in \mathbb{T}$: | For some $n \in \mathbb{Z}, t =n$ |
| Jump operator: | $\sigma(t)=t+1$ |
| Graininess operator: | $\mu(t)=1$ |
| $\Delta$-derivative: | $f^{\Delta}(t)=f(t+1)-f(t)$ |
| $\Delta$-integral: | $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=s}^{t-1} f(k)$ |
| Exponential function: | $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k)) \right) \\ &= \displaystyle\prod_{k=s}^{t-1} \left( 1+p(k) \right) \\ \end{array}$ |
