Integers
From timescalewiki
The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.
| Forward jump: | $\sigma(t)=t+1$ | derivation |
| Forward graininess: | $\mu(t)=1$ | derivation |
| Backward jump: | $\rho(t)=t-1$ | derivation |
| Backward graininess: | $\nu(t)=1$ | derivation |
| $\Delta$-derivative | $f^{\Delta}(t)=f(t+1)-f(t)$ | derivation |
| $\nabla$-derivative | $f^{\nabla}(t)=f(t)-f(t-1)$ | derivation |
| $\Delta$-integral | $\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\int_s^t f(\tau) \Delta \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s}^{t-1} f(k) &; t \gt s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t}^{s-1} f(k) &; t \lt s \end{array} \right.$ | derivation |
| $\nabla$-integral | $\displaystyle\int_s^t f(\tau) \nabla \tau=\displaystyle\int_s^t f(\tau) \nabla \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s+1}^t f(k) &; t>s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t+1}^s f(k) &; t\lt s \end{array} \right.$ | derivation |
| $h_k(t,s)$ | $h_k(t,s)=\dfrac{(t-s)^{\underline{k}}}{k!}$ | derivation |
| $\hat{h}_k(t,s)$ | $\hat{h}_k(t,s)=\dfrac{(t-s)^{\overline{k}}}{k!}$ | derivation |
| $g_k(t,s)$ | $g_k(t,s)=$ | derivation |
| $\hat{g}_k(t,s)$ | $\hat{g}_k(t,s)=$ | derivation |
| $e_p(t,s)$ | $e_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)} &; t < s \\ 1 &; t=s \\ \displaystyle\prod_{k=s}^{t-1} 1+p(k) &; t>s. \end{array} \right.$ | derivation |
| $\hat{e}_p(t,s)$ | $\hat{e}_p(t,s)=$ | derivation |
| Gaussian bell | $\mathbf{E}(t)=2^{\frac{-t(t-1)}{2}}$ | derivation |
| $\mathrm{sin}_p(t,s)$ | $\sin_p(t,s) = \left\{ \begin{array}{ll} \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} &; t>s \\ 0 &; t=s \\ \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i} &; t<s \end{array} \right.$ | derivation |
| $\mathrm{\sin}_1(t,s)$ | $\sin_1(t,s)=$ | derivation |
| $\widehat{\sin}_p(t,s)$ | $\widehat{\sin}_p(t,s)=$ | derivation |
| $\mathrm{\cos}_p(t,s)$ | $\cos_p(t,s)=\begin{array}{ll} \cos_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2} \\ &= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) + \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2} \end{array}$ | derivation |
| $\mathrm{\cos}_1(t,s)$ | $\cos_1(t,s)=\begin{array}{ll} \cos_1(t,0) &= \dfrac{(1+i)^{t}+(1-i)^{t}}{2} \\ &= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k + \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2} \end{array}$ | derivation |
| $\widehat{\cos}_p(t,s)$ | $\widehat{\cos}_p(t,s)=$ | derivation |
| $\sinh_p(t,s)$ | $\sinh_p(t,s)=$ | derivation |
| $\widehat{\sinh}_p(t,s)$ | $\widehat{\sinh}_p(t,s)=$ | derivation |
| $\cosh_p(t,s)$ | $\cosh_p(t,s)=$ | derivation |
| $\widehat{\cosh}_p(t,s)$ | $\widehat{\cosh}_p(t,s)=$ | derivation |
| Gamma function | $\Gamma_{\mathbb{Z}}(x,s)=\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{2^{k+1}}$ | derivation |
| Euler-Cauchy logarithm | $L(t,s)=$ | derivation |
| Bohner logarithm | $L_p(t,s)=$ | derivation |
| Jackson logarithm | $\log_{\mathbb{Z}} g(t)=$ | derivation |
| Mozyrska-Torres logarithm | $L_{\mathbb{Z}}(t)=$ | derivation |
| Laplace transform | $\mathscr{L}_{\mathbb{Z}}\{f\}(z;s)=$ | derivation |
| Hilger circle | 
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derivation |
