Difference between revisions of "Real numbers"

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[[Gaussian bell]]
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|[[Derivation of Gaussian bell for T=R|derivation]]
 
|[[Derivation of Gaussian bell for T=R|derivation]]

Revision as of 23:55, 21 May 2015

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: $\sigma(t)=t$ 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
Gaussian bell 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