Integers

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The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.

$\mathbb{T}=\mathbb{Z}$
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: $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k)) \right) \\ &= \displaystyle\prod_{k=s}^{t-1} \left( 1+p(k) \right) \\ \end{array}$
$\mathrm{sin}_p(t,0)$ $$\begin{array}{ll} \sin_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i} \\ &= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i} \end{array}$$
$\mathrm{sin}_1(t,0)$ $$\begin{array}{ll} \sin_1(t,0) &= \dfrac{(1+i)^{t}-(1-i)^{t}}{2i} \\ &= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k - \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2i} \end{array}$$

Sin 1T=Z.png

$\mathrm{cos}_p(t,t_0)$ $$\begin{array}{ll} \cos_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2} \\ &= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k + \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2} \\ \end{array}$$

Cos 1T=Z.png