Difference between revisions of "Nth root of nonnegative integers"
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| − | The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a [[time scale]]. | + | Let $n \in \{1,2,\ldots\}$. The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a [[time scale]]. |
{| class="wikitable" | {| class="wikitable" | ||
| − | |+$\mathbb{T}=\sqrt[n]{\mathbb{N}}$ | + | |+$\mathbb{T}=\sqrt[n]{\mathbb{N}_0}$ |
|- | |- | ||
|[[Forward jump]]: | |[[Forward jump]]: | ||
| − | |$\sigma(t)=$ | + | |$\sigma(t)=\sqrt[n]{t^n+1}$ |
|[[Derivation of forward jump for T=nth root of nonnegative integers|derivation]] | |[[Derivation of forward jump for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Forward graininess]]: | |[[Forward graininess]]: | ||
| − | |$\mu(t)=$ | + | |$\mu(t)=\sqrt[n]{t^n+1}-t$ |
|[[Derivation of forward graininess for T=nth root of nonnegative integers|derivation]] | |[[Derivation of forward graininess for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Backward jump]]: | |[[Backward jump]]: | ||
| − | |$\rho(t)=$ | + | |$\rho(t)=\sqrt[n]{t^n-1}$ |
|[[Derivation of backward jump for T=nth root of nonnegative integers|derivation]] | |[[Derivation of backward jump for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Backward graininess]]: | |[[Backward graininess]]: | ||
| − | |$\nu(t)=$ | + | |$\nu(t)=t-\sqrt[n]{t^n-1}$ |
|[[Derivation of backward graininess for T=nth root of nonnegative integers|derivation]] | |[[Derivation of backward graininess for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Delta derivative | $\Delta$-derivative]] | |[[Delta derivative | $\Delta$-derivative]] | ||
| − | |$f^{\Delta}(t)=$ | + | |$f^{\Delta}(t)=\dfrac{f(\sqrt[n]{t^n+1})-f(t)}{\sqrt[n]{t^n+1}-t}$ |
|[[Derivation of delta derivative for T=nth root of nonnegative integers|derivation]] | |[[Derivation of delta derivative for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Nabla derivative | $\nabla$-derivative]] | |[[Nabla derivative | $\nabla$-derivative]] | ||
| − | |$f^{\nabla}(t)=$ | + | |$f^{\nabla}(t)=\dfrac{f(t)-f(\sqrt[n]{t^n-1})}{t-\sqrt{t^n-1}}$ |
|[[Derivation of nabla derivative for T=nth root of nonnegative integers|derivation]] | |[[Derivation of nabla derivative for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
|[[Delta integral | $\Delta$-integral]] | |[[Delta integral | $\Delta$-integral]] | ||
| − | |$\displaystyle\int_s^t f(\tau) \Delta \tau=$ | + | |$\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\sum_{k=s^n}^{t^n-1} (\sqrt[n]{k+1}-\sqrt[n]{k}) f(\sqrt[n]{k})$ |
|[[Derivation of delta integral for T=nth root of nonnegative integers|derivation]] | |[[Derivation of delta integral for T=nth root of nonnegative integers|derivation]] | ||
|- | |- | ||
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|- | |- | ||
|} | |} | ||
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| + | <center>{{:Time scales footer}}</center> | ||
Latest revision as of 00:56, 11 December 2016
Let $n \in \{1,2,\ldots\}$. The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a time scale.
| Forward jump: | $\sigma(t)=\sqrt[n]{t^n+1}$ | derivation |
| Forward graininess: | $\mu(t)=\sqrt[n]{t^n+1}-t$ | derivation |
| Backward jump: | $\rho(t)=\sqrt[n]{t^n-1}$ | derivation |
| Backward graininess: | $\nu(t)=t-\sqrt[n]{t^n-1}$ | derivation |
| $\Delta$-derivative | $f^{\Delta}(t)=\dfrac{f(\sqrt[n]{t^n+1})-f(t)}{\sqrt[n]{t^n+1}-t}$ | derivation |
| $\nabla$-derivative | $f^{\nabla}(t)=\dfrac{f(t)-f(\sqrt[n]{t^n-1})}{t-\sqrt{t^n-1}}$ | derivation |
| $\Delta$-integral | $\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\sum_{k=s^n}^{t^n-1} (\sqrt[n]{k+1}-\sqrt[n]{k}) f(\sqrt[n]{k})$ | derivation |
| $\nabla$-integral | $\displaystyle\int_s^t f(\tau) \nabla \tau=$ | derivation |
| $h_k(t,s)$ | $h_k(t,s)=$ | derivation |
| $\hat{h}_k(t,s)$ | $\hat{h}_k(t,s)=$ | derivation |
| $g_k(t,s)$ | $g_k(t,s)=$ | derivation |
| $\hat{g}_k(t,s)$ | $\hat{g}_k(t,s)=$ | derivation |
| $e_p(t,s)$ | $e_p(t,s)=$ | derivation |
| $\hat{e}_p(t,s)$ | $\hat{e}_p(t,s)=$ | derivation |
| Gaussian bell | $\mathbf{E}(t)=$ | derivation |
| $\mathrm{sin}_p(t,s)=$ | $\sin_p(t,s)=$ | derivation |
| $\mathrm{\sin}_1(t,s)$ | $\sin_1(t,s)=$ | derivation |
| $\widehat{\sin}_p(t,s)$ | $\widehat{\sin}_p(t,s)=$ | derivation |
| $\mathrm{\cos}_p(t,s)$ | $\cos_p(t,s)=$ | derivation |
| $\mathrm{\cos}_1(t,s)$ | $\cos_1(t,s)=$ | 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 | $\Gamma_{\sqrt[n]{\mathbb{N}}}(x,s)=$ | derivation |
| Euler-Cauchy logarithm | $L(t,s)=$ | derivation |
| Bohner logarithm | $L_p(t,s)=$ | derivation |
| Jackson logarithm | $\log_{\sqrt[n]{\mathbb{N}}} g(t)=$ | derivation |
| Mozyrska-Torres logarithm | $L_{\sqrt[n]{\mathbb{N}}}(t)=$ | derivation |
| Laplace transform | $\mathscr{L}_{\sqrt[n]{\mathbb{N}}}\{f\}(z;s)=$ | derivation |
| Hilger circle | derivation |
