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.
| Forward jump: | $\sigma(t)=t$ | derivation |
| Forward graininess: | $\mu(t)=0$ | derivation |
| Backward jump: | $\rho(t)=t$ | derivation |
| Backward graininess: | $\nu(t)=0$ | derivation |
| $\Delta$-derivative | $f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$ | derivation |
| $\nabla$-derivative | $f^{\nabla}(t) =\displaystyle\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$ | derivation |
| $\Delta$-integral | $\displaystyle\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$ | derivation |
| $\nabla$-derivative | $\displaystyle\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ | derivation |
| $h_k(t,s)$ | derivation | |
| $\hat{h}_k(t,s)$ | derivation | |
| $g_k(t,s)$ | derivation | |
| $\hat{g}_k(t,s)$ | derivation | |
| $e_p(t,s)=$ | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | derivation |
| $\hat{e}_p(t,s)=$ | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | derivation |
| derivation | ||
| $\mathrm{sin}_p(t,s)=$ | $\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | derivation |
| $\mathrm{\sin}_1(t,0)$ | $\sin(t)$ | derivation |
| $\widehat{\sin}_p(t,s)$ | derivation | |
| $\mathrm{\cos}_p(t,s)$ | $\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | derivation |
| $\mathrm{\cos}_1(t,0)$ | $\cos(t)$ | derivation |
| $\widehat{\cos}_p(t,s)$ | derivation | |
| $\sinh_p(t,s)$ | derivation | |
| $\widehat{\sinh}_p(t,s)$ | derivation | |
| $\cosh_p(t,s)$ | derivation | |
| $\widehat{\cosh}_p(t,s)$ | derivation | |
| Gamma function | $\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ | derivation |
| Euler-Cauchy logarithm | derivation | |
| Bohner logarithm | derivation | |
| Jackson logarithm | derivation | |
| Mozyrska-Torres logarithm | derivation | |
| Laplace transform | $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ | derivation |
| Hilger circle | derivation |
