Latest revision as of 15:55, 15 January 2023

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

$\mathbb{T}=h\mathbb{Z}$

Forward jump:	$\sigma(t)=t+h$	derivation
Forward graininess:	$\mu(t)=h$	derivation
Backward jump:	$\rho(t)=t-h$	derivation
Backward graininess:	$\nu(t)=h$	derivation
$\Delta$-derivative	$f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$	derivation
$\nabla$-derivative	$f^{\nabla}(t)=\dfrac{f(t)-f(t-h)}{h}$	derivation
$\Delta$-integral	$\displaystyle\int_s^t f(\tau) \Delta \tau=$	derivation
$\nabla$-integral	$\displaystyle\int_s^t f(\tau) \nabla \tau=$	derivation
$h_k(t,s)$	$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$	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
$e_p(t,s)$	$e_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s \\ 1 &; t=s \\ \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} 1+hp(hk) &; t>s. \end{array} \right.$	derivation
$\hat{e}_p(t,s)$	$\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{t}{h}+1}^{\frac{s}{h}} (1-hp(hk)) &; t \lt s \\ 1 &; t=s \\ \displaystyle\prod_{k=\frac{s}{h}+1}^{\frac{t}{h}} \dfrac{1}{1-hp(hk)} &; t \gt s. \end{array} \right.$	derivation
Gaussian bell	$\mathbf{E}(t)=\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$	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	$\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$	derivation
Euler-Cauchy logarithm	$L(t,s)=$	derivation
Bohner logarithm	$L_p(t,s)=$	derivation
Jackson logarithm	$\log_{h\mathbb{Z}} g(t)=$	derivation
Mozyrska-Torres logarithm	$L_{h\mathbb{Z}}(t)=$	derivation
Laplace transform	$\mathscr{L}_{h\mathbb{Z}}\{f\}(z;s)=$	derivation
Hilger circle		derivation

References

• Robert J. Marks II, Ian A. Gravagne and John M. Davis: A generalized Fourier transform and convolution on time scales (2008)... (previous): Section 2.1(a)
• Billy Jackson: Partial dynamic equations on time scales (2006)... (previous)... (next): Appendix

Cauchy Functions and Taylor's Formula for Time Scales T