Difference between revisions of "Real numbers"
From timescalewiki
| Line 15: | Line 15: | ||
|$\Delta$-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: | ||
| + | | $\begin{array}{ll} | ||
| + | e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ | ||
| + | &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ | ||
| + | &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) | ||
| + | \end{array}$ | ||
|} | |} | ||
Revision as of 03:35, 18 May 2014
The set $\mathbb{R}$ of real numbers is a time scale.
| 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: | $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$ |
