Difference between revisions of "Real numbers"
From timescalewiki
| Line 6: | Line 6: | ||
|[[Forward jump]]: | |[[Forward jump]]: | ||
|$\sigma(t)=t$ | |$\sigma(t)=t$ | ||
| + | |[[Derivation of forward jump for T=R|derivation]] | ||
|- | |- | ||
|[[Forward graininess]]: | |[[Forward graininess]]: | ||
|$\mu(t)=0$ | |$\mu(t)=0$ | ||
| + | |[[Derivation of forward graininess for T=R|derivation]] | ||
|- | |- | ||
|[[Backward jump]]: | |[[Backward jump]]: | ||
|$\rho(t)=t$ | |$\rho(t)=t$ | ||
| + | |[[Derivation of backward jump for T=R|derivation]] | ||
|- | |- | ||
|[[Backward graininess]]: | |[[Backward graininess]]: | ||
|$\nu(t)=0$ | |$\nu(t)=0$ | ||
| + | |[[Derivation of backward graininess for T=R|derivation]] | ||
|- | |- | ||
|[[Delta derivative | $\Delta$-derivative]] | |[[Delta derivative | $\Delta$-derivative]] | ||
| − | |$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$ | + | |$f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$ |
| + | |[[Derivation of delta derivative for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla derivative | $\nabla$-derivative]] | |[[Nabla derivative | $\nabla$-derivative]] | ||
| − | |$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$ | + | |$f^{\nabla}(t) =\displaystyle\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$ |
| + | |[[Derivation of nabla derivative for T=R|derivation]] | ||
|- | |- | ||
|[[Delta integral | $\Delta$-integral]] | |[[Delta integral | $\Delta$-integral]] | ||
| − | | $\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$ | + | |$\displaystyle\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$ |
| + | |[[Derivation of delta integral for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla derivative | $\nabla$-derivative]] | |[[Nabla derivative | $\nabla$-derivative]] | ||
| − | |$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ | + | |$\displaystyle\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ |
| + | |[[Derivation of nabla integral for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Delta hk|$h_k(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of delta hk for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Nabla hk|$\hat{h}_k(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of nabla hk for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Delta gk|$g_k(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of delta gk for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Nabla gk|$\hat{g}_k(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of nabla gk for T=R|derivation]] | ||
|- | |- | ||
|[[Delta exponential | $e_p(t,s)=$]] | |[[Delta exponential | $e_p(t,s)=$]] | ||
| − | | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| − | + | |[[Derivation of delta exponential T=R|derivation]] | |
|- | |- | ||
|[[Nabla exponential | $\hat{e}_p(t,s)=$]] | |[[Nabla exponential | $\hat{e}_p(t,s)=$]] | ||
|$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | |$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | ||
| − | + | |[[Derivation of nabla exponential T=R|derivation]] | |
|- | |- | ||
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | ||
|$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | |$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | ||
| − | + | |[[Derivation of delta sin sub p for T=R|derivation]] | |
|- | |- | ||
|$\mathrm{\sin}_1(t,0)$ | |$\mathrm{\sin}_1(t,0)$ | ||
|$\sin(t)$ | |$\sin(t)$ | ||
| − | + | |[[Derivation of delta sin sub 1 for T=R|derivation]] | |
|- | |- | ||
| − | |$\mathrm{\cos}_p(t,s)$ | + | |[[Nabla sine|$\widehat{\sin}_p(t,s)$]] |
| + | | | ||
| + | |[[Derivation of nabla sine sub p for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Delta cosine|$\mathrm{\cos}_p(t,s)$]] | ||
|$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | ||
| − | + | |[[Derivation of delta cos sub p for T=R|derivation]] | |
|- | |- | ||
|$\mathrm{\cos}_1(t,0)$ | |$\mathrm{\cos}_1(t,0)$ | ||
|$\cos(t)$ | |$\cos(t)$ | ||
| − | ([[Derivation of cos sub 1 for T=R|derivation]]) | + | |[[Derivation of delta cos sub 1 for T=R|derivation]] |
| + | |- | ||
| + | |[[Nabla cosine|$\widehat{\cos}_p(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of nabla cos sub 1 for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Delta sinh|$\sinh_p(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of delta sinh sub p for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of nabla sinh sub p for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Delta cosh|$\cosh_p(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of delta cosh sub p for T=R|derivation]] | ||
| + | |- | ||
| + | |[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]] | ||
| + | | | ||
| + | |[[Derivation of nabla cosh sub p for T=R|derivation]] | ||
|- | |- | ||
|[[Hilger circle]] | |[[Hilger circle]] | ||
| | | | ||
| + | |[[Derivation of Hilger circle for T=R|derivation]] | ||
|- | |- | ||
|[[Laplace transform]] | |[[Laplace transform]] | ||
|$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ | |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ | ||
| + | |[[Derivation of Laplace transform for T=R|derivation]] | ||
|- | |- | ||
|[[Gamma function]] | |[[Gamma function]] | ||
|$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ | |$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ | ||
| + | |[[Derivation of gamma function for T=R|derivation]] | ||
|- | |- | ||
|} | |} | ||
Revision as of 23:46, 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.
| 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 |
| $\mathrm{sin}_p(t,s)=$ | $\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | 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 | |
| Hilger circle | derivation | |
| Laplace transform | $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ | derivation |
| Gamma function | $\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ | derivation |
