# Real numbers

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The set $\mathbb{R}$ of real numbers is a time scale. In this time scale, all derivatives reduce to the classical derivative and the integrals reduce to the classical 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 $h_k(t,s)$ $h_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation $\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation $g_k(t,s)$ $g_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation $\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation $e_p(t,s)$ $e_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation $\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation Gaussian bell $\mathbf{E}(t)=e^{-\frac{t^2}{2}}$ derivation $\mathrm{sin}_p(t,s)=$ $\sin_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation $\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=\sin(t-s)$ derivation $\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation $\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation $\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=\cos(t-s)$ derivation $\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation $\sinh_p(t,s)$ $\sinh_p(t,s)=$ derivation $\widehat{\sinh}_p(t,s)$ $\widehat{\sinh}_p(t,s)=$ derivation $\cosh_p(t,s)$ $\cosh_p(t,s)=$ derivation $\widehat{\cosh}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ 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 $L(t,s)=$ derivation Bohner logarithm $L_p(t,s)=$ derivation Jackson logarithm $\log_{\mathbb{T}} g(t)=$ derivation Mozyrska-Torres logarithm $L_{\mathbb{T}}(t)=$ 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