Revision as of 22:16, 23 February 2016
The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a time scale.
$\mathbb{T}=\sqrt[n]{\mathbb{N}}$
| Forward jump:
|
$\sigma(t)=$
|
derivation
|
| Forward graininess:
|
$\mu(t)=$
|
derivation
|
| Backward jump:
|
$\rho(t)=$
|
derivation
|
| Backward graininess:
|
$\nu(t)=$
|
derivation
|
| $\Delta$-derivative
|
$f^{\Delta}(t)=$
|
derivation
|
| $\nabla$-derivative
|
$f^{\nabla}(t)=$
|
derivation
|
| $\Delta$-integral
|
$\displaystyle\int_s^t f(\tau) \Delta \tau=$
|
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
|
