Real numbers

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.

$\mathbb{T}=\mathbb{R}$

Forward jump:	$\sigma(t)=t$
Forward 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	$$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$
$e_p(t,s)=$	$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ (derivation)
$\hat{e}_p(t,s)=$	$$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of nabla exponential T=R|derivation]]) |- |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] |$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of sin sub p for T=R|derivation]]) |- |$\mathrm{\sin}_1(t,0)$ |$\sin(t)$ ([[Derivation of sin sub 1 for T=R|derivation]]) |- |$\mathrm{\cos}_p(t,s)$ |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of cos sub p for T=R|derivation]]) |- |$\mathrm{\cos}_1(t,0)$ |$\cos(t)$ ([[Derivation of cos sub 1 for T=R|derivation]]) |- |[[Hilger circle]] | |- |[[Laplace transform]] |$$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$$ |- |[[Gamma function]] |$$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$$