Difference between revisions of "Real numbers"
From timescalewiki
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|[[Delta exponential | $e_p(t,s)=$]] | |[[Delta exponential | $e_p(t,s)=$]] | ||
| − | | | + | | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
([[Derivation of delta exponential T=R|derivation]]) | ([[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]]) | ([[Derivation of nabla exponential T=R|derivation]]) | ||
|- | |- | ||
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | ||
| − | | | + | |$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
([[Derivation of sin sub p for T=R|derivation]]) | ([[Derivation of sin sub p for T=R|derivation]]) | ||
|- | |- | ||
|$\mathrm{\sin}_1(t,0)$ | |$\mathrm{\sin}_1(t,0)$ | ||
| − | | | + | |$\sin(t)$ |
([[Derivation of sin sub 1 for T=R|derivation]]) | ([[Derivation of sin sub 1 for T=R|derivation]]) | ||
|- | |- | ||
|$\mathrm{\cos}_p(t,s)$ | |$\mathrm{\cos}_p(t,s)$ | ||
| − | | | + | |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
([[Derivation of cos sub p for T=R|derivation]]) | ([[Derivation of cos sub p for T=R|derivation]]) | ||
|- | |- | ||
Revision as of 19:36, 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 | $$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ |
| $e_p(t,s)=$ | $\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
| $\hat{e}_p(t,s)=$ | $$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of nabla exponential T=R|derivation]]) |- |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] |$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of sin sub p for T=R|derivation]]) |- |$\mathrm{\sin}_1(t,0)$ |$\sin(t)$ ([[Derivation of sin sub 1 for T=R|derivation]]) |- |$\mathrm{\cos}_p(t,s)$ |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ ([[Derivation of cos sub p for T=R|derivation]]) |- |$\mathrm{\cos}_1(t,0)$ |$\cos(t)$ ([[Derivation of cos sub 1 for T=R|derivation]]) |- |[[Hilger circle]] | |- |[[Laplace transform]] |$$\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$$ |
