Real numbers
From timescalewiki
The set $\mathbb{R}$ of real numbers is a time scale.
| Generic element $t \in \mathbb{T}$: | $t=t$ |
| 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: | $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$ |
