Difference between revisions of "Integers"

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{| class="wikitable"
 
{| class="wikitable"
−
|+$\mathbb{T}=h\mathbb{Z}$
+
|+$\mathbb{T}=\mathbb{Z}$
 
|-
 
|-
−
|Generic element $t\in \mathbb{T}$:
+
|[[Forward jump]]:
−
|For some $n \in \mathbb{Z}, t =n$
 
−
|-
 
−
|Jump operator:
 
 
|$\sigma(t)=t+1$
 
|$\sigma(t)=t+1$
 +
|[[Derivation of forward jump for T=Z|derivation]]
 
|-
 
|-
−
|Graininess operator:
+
|[[Forward graininess]]:
 
|$\mu(t)=1$
 
|$\mu(t)=1$
 +
|[[Derivation of forward graininess for T=Z|derivation]]
 +
|-
 +
|[[Backward jump]]:
 +
|$\rho(t)=t-1$
 +
|[[Derivation of backward jump for T=Z|derivation]]
 +
|-
 +
|[[Backward graininess]]:
 +
|$\nu(t)=1$
 +
|[[Derivation of backward graininess for T=Z|derivation]]
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
+
|[[Delta derivative | $\Delta$-derivative]]
 
|$f^{\Delta}(t)=f(t+1)-f(t)$
 
|$f^{\Delta}(t)=f(t+1)-f(t)$
 +
|[[Derivation of delta derivative for T=Z|derivation]]
 +
|-
 +
|[[Nabla derivative | $\nabla$-derivative]]
 +
|$f^{\nabla}(t)=f(t)-f(t-1)$
 +
|[[Derivation of nabla derivative for T=Z|derivation]]
 +
|-
 +
|[[Delta integral | $\Delta$-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 of delta integral for T=Z|derivation]]
 +
|-
 +
|[[Nabla integral | $\nabla$-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 of nabla integral for T=Z|derivation]]
 +
|-
 +
|[[Delta hk|$h_k(t,s)$]]
 +
|$h_k(t,s)=\dfrac{(t-s)^{\underline{k}}}{k!}$
 +
|[[Derivation of delta hk for T=Z|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=\dfrac{(t-s)^{\overline{k}}}{k!}$
 +
|[[Derivation of nabla hk for T=Z|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=$
 +
|[[Derivation of delta gk for T=Z|derivation]]
 +
|-
 +
|[[Nabla gk|$\hat{g}_k(t,s)$]]
 +
|$\hat{g}_k(t,s)=$
 +
|[[Derivation of nabla gk for T=Z|derivation]]
 +
|-
 +
|[[Delta exponential | $e_p(t,s)$]]
 +
|$e_p(t,s)=\left\{ \begin{array}{ll}
 +
\displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)} &; t < s \\
 +
1 &; t=s \\
 +
\displaystyle\prod_{k=s}^{t-1} 1+p(k) &; t>s.
 +
\end{array} \right.$
 +
|[[Derivation of delta exponential T=Z|derivation]]
 +
|-
 +
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
 +
|$\hat{e}_p(t,s)=$
 +
|[[Derivation of nabla exponential T=Z|derivation]]
 +
|-
 +
|[[Gaussian bell]]
 +
|$\mathbf{E}(t)=2^{\frac{-t(t-1)}{2}}$
 +
|[[Derivation of Gaussian bell for T=Z|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)$]]
 +
|$\sin_p(t,s) = \left\{ \begin{array}{ll}
 +
\dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} &; t>s \\
 +
0 &; t=s \\
 +
\dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i} &; t<s
 +
\end{array} \right.$
 +
|[[Derivation of delta sin sub p for T=Z|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=$
 +
|[[Derivation of delta sin sub 1 for T=Z|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=$
 +
|[[Derivation of nabla sine sub p for T=Z|derivation]]
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
+
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
−
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=s}^{t-1} f(k)$
+
|$\cos_p(t,s)=\begin{array}{ll}
 +
\cos_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2} \\
 +
&= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) + \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2}
 +
\end{array}$
 +
|[[Derivation of delta cos sub p for T=Z|derivation]]
 
|-
 
|-
−
|[[Exponential_functions | Exponential function]]:
+
|$\mathrm{\cos}_1(t,s)$
−
| $\begin{array}{ll}
+
|$\cos_1(t,s)=\begin{array}{ll}
−
e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\
+
\cos_1(t,0) &= \dfrac{(1+i)^{t}+(1-i)^{t}}{2} \\
−
&= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k))  \right) \\
+
&= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k + \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2}
−
&= \displaystyle\prod_{k=s}^{t-1} \left( 1+p(k) \right) \\
 
 
\end{array}$
 
\end{array}$
 +
|[[Derivation of delta cos sub 1 for T=Z|derivation]]
 +
|-
 +
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cos sub 1 for T=Z|derivation]]
 +
|-
 +
|[[Delta sinh|$\sinh_p(t,s)$]]
 +
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=Z|derivation]]
 +
|-
 +
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
 +
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=Z|derivation]]
 +
|-
 +
|[[Delta cosh|$\cosh_p(t,s)$]]
 +
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=Z|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cosh}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=Z|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 of gamma function for T=Z|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=Z|derivation]]
 +
|-
 +
|[[Bohner logarithm]]
 +
|$L_p(t,s)=$
 +
|[[Derivation of the Bohner logarithm for T=Z|derivation]]
 +
|-
 +
|[[Jackson logarithm]]
 +
|$\log_{\mathbb{Z}} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=Z|derivation]]
 +
|-
 +
|[[Mozyrska-Torres logarithm]]
 +
|$L_{\mathbb{Z}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=Z|derivation]]
 +
|-
 +
|[[Laplace transform]]
 +
|$\mathscr{L}_{\mathbb{Z}}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=Z|derivation]]
 +
|-
 +
|[[Hilger circle]]
 +
|[[File:Hilgercircle,T=Z.png|250px]]
 +
|[[Derivation of Hilger circle for T=Z|derivation]]
 +
|-
 
|}
 
|}
 +
 +
<center>{{:Time scales footer}}</center>

Latest revision as of 01:14, 19 February 2016

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

$\mathbb{T}=\mathbb{Z}$
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 Hilgercircle,T=Z.png derivation

Examples of time scales
$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set
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