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