Difference between revisions of "Real numbers"
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|- | |- | ||
|[[Delta derivative | $\Delta$-derivative]] | |[[Delta derivative | $\Delta$-derivative]] | ||
| − | | | + | |$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$ |
|- | |- | ||
|[[Nabla derivative | $\nabla$-derivative]] | |[[Nabla derivative | $\nabla$-derivative]] | ||
| − | | | + | |$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$ |
|- | |- | ||
|[[Delta integral | $\Delta$-integral]] | |[[Delta integral | $\Delta$-integral]] | ||
| − | | | + | | $\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$ |
|- | |- | ||
|[[Nabla derivative | $\nabla$-derivative]] | |[[Nabla derivative | $\nabla$-derivative]] | ||
| − | | | + | |$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ |
|- | |- | ||
|[[Delta exponential | $e_p(t,s)=$]] | |[[Delta exponential | $e_p(t,s)=$]] | ||
| Line 33: | Line 33: | ||
|- | |- | ||
|[[Nabla exponential | $\hat{e}_p(t,s)=$]] | |[[Nabla exponential | $\hat{e}_p(t,s)=$]] | ||
| − | | | + | |$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
([[Derivation of nabla exponential T=R|derivation]]) | ([[Derivation of nabla exponential T=R|derivation]]) | ||
|- | |- | ||
| Line 56: | Line 56: | ||
|- | |- | ||
|[[Laplace transform]] | |[[Laplace transform]] | ||
| − | | | + | |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ |
|- | |- | ||
|[[Gamma function]] | |[[Gamma function]] | ||
| − | | | + | |$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ |
|- | |- | ||
|} | |} | ||
Revision as of 19:38, 29 April 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.
| 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 | $\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ |
| $e_p(t,s)=$ | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\hat{e}_p(t,s)=$ | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\mathrm{sin}_p(t,s)=$ | $\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\mathrm{\sin}_1(t,0)$ | $\sin(t)$ |
| $\mathrm{\cos}_p(t,s)$ | $\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\mathrm{\cos}_1(t,0)$ | $\cos(t)$ |
| Hilger circle | |
| Laplace transform | $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ |
| Gamma function | $\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ |
