Difference between revisions of "Multiples of integers"

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|[[Delta_integral | $\Delta$-integral:]]
 
|[[Delta_integral | $\Delta$-integral:]]
 
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} hf(hk)$
 
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} hf(hk)$
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|-
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|[[Polynomials | $h_k(t,s)$]]
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|$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
 
|-
 
|-
 
|[[Exponential_functions | Exponential function]]:
 
|[[Exponential_functions | Exponential function]]:

Revision as of 05:13, 2 November 2014

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

$\mathbb{T}=h\mathbb{Z}$
Generic element $t\in \mathbb{T}$: For some $n \in \mathbb{Z}, t =hn$
Jump operator: $\sigma(t)=t+h$
Graininess operator: $\mu(t)=h$
$\Delta$-derivative: $f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$
$\Delta$-integral: $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} hf(hk)$
$h_k(t,s)$ $h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
Exponential function: $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + hp(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} \log(1+hp(hk)) \right) \\ &= \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \left( 1+hp(hk) \right) \\ \end{array}$
Hilger circle: Hilgercircle,T=hZ.png
Gamma function: $\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$

References

Cauchy Functions and Taylor's Formula for Time Scales T