Difference between revisions of "Real numbers"

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|$\mu(t)=0$
 
|$\mu(t)=0$
 
|-
 
|-
|$\Delta$-derivative:
+
|[[Delta_derivative | $\Delta$-derivative:]]
 
|$f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}$
 
|$f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}$
 
|-
 
|-
|$\Delta$-integral:
+
|[[Delta_integral | $\Delta$-integral:]]
 
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\int_s^t f(\tau) d\tau$ is the Riemann integral
 
| $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\int_s^t f(\tau) d\tau$ is the Riemann integral
 
|-
 
|-
|Exponential function:
+
|[[Exponential_functions | Exponential function]]:
 
| $\begin{array}{ll}
 
| $\begin{array}{ll}
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\
 
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\

Revision as of 03:37, 18 May 2014

The set $\mathbb{R}$ of real numbers is a time scale.

$\mathbb{T}=\mathbb{R}$
Jump operator: $\sigma(t)=t$
Graininess operator: $\mu(t)=0$
$\Delta$-derivative: $f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}$
$\Delta$-integral: $\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\int_s^t f(\tau) d\tau$ is the Riemann integral
Exponential function: $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$