Real numbers
From timescalewiki
The set $\mathbb{R}$ of real numbers is a time scale. In this time scale, all derivatives reduce to the clasical derivative and the integrals reduce to the Riemann integral.
| Generic element $t \in \mathbb{T}$: | $t=t$ |
| Forward jump: | $\sigma(t)=t$ |
| Graininess: | $\mu(t)=0$ |
| Backward jump: | $\rho(t)=t$ |
| Backward graininess: | $\nu(t)=0$ |
| $\Delta$-derivative | $$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$$ |
| $\nabla$-derivative | $$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$$ |
| $\Delta$-integral | $$\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$$ |
| $\nabla$-derivative | $$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ |
| $\Delta$-exponential | $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$ |
| $\nabla$-exponential | $\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\mathrm{sin}_p(t,s)$ | |
| $\mathrm{\sin}_1(t,0)$ | $\sin(t)$ |
| $\mathrm{\cos}_p(t,0)$ | $\cos \left( \displaystyle\int_0^t p(\tau) d\tau \right)$ |
| $\mathrm{\cos}_1(t,0)$ | $\cos(t)$ |
| Hilger circle | |
| Laplace transform | $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ |
| Gamma function |
