Integers
From timescalewiki
The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.
| Forward jump: | $\sigma(t)=t+1$ |
| Forward graininess: | $\mu(t)=1$ |
| Backward jump: | $\rho(t)=t-1$ |
| Backward graininess: | $\nu(t)=1$ |
| $\Delta$-derivative | $f^{\Delta}(t)=f(t+1)-f(t)$ |
| $\nabla$-derivative | $f^{\nabla}(t)=f(t)-f(t-1)$ |
| $\Delta$-integral | $\int_s^t f(\tau) \Delta \tau = \left\{ \begin{array}{ll} \sum_{k=s}^{t-1} f(k) &; t \gt s \\ 0 &; t=s \\ -\sum_{k=t}^{s-1} f(k) &; t \lt s \end{array} \right.$ |
| $\nabla$-integral | $\int_s^t f(\tau) \nabla \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s+1}^t f(k) &; t>s \\ 0 &; t=s \\ -\sum_{k=t+1}^s f(k) &; t\lt s \end{array} \right.$ |
| $\Delta$-exponential | $e_p(t,s) = \left\{ \begin{array}{ll} \displaystyle\prod_{k=s}^{t-1} 1+p(k) &= t \gt s \\ 1 &= t=s \\ \displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)} &= t \gt s \\ \end{array} \right.$ |
| $\nabla$-exponential | |
| $\mathrm{sin}_p(t,s)$ | $\sin_p(t,s) = \left\{ \begin{array}{ll}
\dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} &; t>s \\
0 &; t=s \\
\dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i} &; t<s
\end{array} \right.$ |
| $\mathrm{sin}_1(t,0)$ | $\begin{array}{ll}
\sin_1(t,0) &= \dfrac{(1+i)^{t}-(1-i)^{t}}{2i} \\
&= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k - \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2i}
\end{array}$
|
| $\mathrm{cos}_p(t,t_0)$ | $\begin{array}{ll} \cos_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2} \\ &= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) + \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2} \end{array}$ |
| $\mathrm{cos}_1(t,0)$ | \begin{array}{ll}
\cos_1(t,0) &= \dfrac{(1+i)^{t}+(1-i)^{t}}{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}
|
| Hilger circle | 
|
| Gamma function: | $\Gamma_{\mathbb{Z}}(t;s)=\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{2^{k+1}}$ |
