Difference between revisions of "Real numbers"
From timescalewiki
| Line 3: | Line 3: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+$\mathbb{T}=\mathbb{R}$ | |+$\mathbb{T}=\mathbb{R}$ | ||
| − | |||
| − | |||
| − | |||
|- | |- | ||
|[[Forward jump]]: | |[[Forward jump]]: | ||
|$\sigma(t)=t$ | |$\sigma(t)=t$ | ||
|- | |- | ||
| − | |[[ | + | |[[Forward graininess]]: |
|$\mu(t)=0$ | |$\mu(t)=0$ | ||
|- | |- | ||
| Line 31: | Line 28: | ||
|$$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ | |$$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ | ||
|- | |- | ||
| − | |[[Delta exponential | $ | + | |[[Delta exponential | $e_p(t,s)=$]] |
| − | | | + | | $$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$$ |
| + | ([[Derivation of delta exponential T=R|derivation]]) | ||
|- | |- | ||
| − | |[[Nabla exponential | + | |[[Nabla exponential | $\hat{e}_p(t,s)=$]] |
| − | |$\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$$ |
| + | ([[Derivation of nabla exponential T=R|derivation]]) | ||
|- | |- | ||
| − | |[[ | + | |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] |
| + | |$$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$$ | ||
| + | ([[Derivation of sin sub p for T=R|derivation]]) | ||
|- | |- | ||
|$\mathrm{\sin}_1(t,0)$ | |$\mathrm{\sin}_1(t,0)$ | ||
| − | |$\sin(t)$ | + | |$$\sin(t)$$ |
| + | ([[Derivation of sin sub 1 for T=R|derivation]]) | ||
|- | |- | ||
|$\mathrm{\cos}_p(t,s)$ | |$\mathrm{\cos}_p(t,s)$ | ||
| − | |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$$ |
| + | ([[Derivation of 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]]) | ||
|- | |- | ||
|[[Hilger circle]] | |[[Hilger circle]] | ||
| Line 52: | Line 56: | ||
|- | |- | ||
|[[Laplace transform]] | |[[Laplace transform]] | ||
| − | |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ | + | |$$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$$ |
|- | |- | ||
|[[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 19:26, 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 | $$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$$ |
