Difference between revisions of "Real numbers"
From timescalewiki
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| − | The set $\mathbb{R}$ of real numbers is a [[time scale]]. | + | 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 [http://en.wikipedia.org/wiki/Riemann_integral Riemann integral]. |
{| class="wikitable" | {| class="wikitable" | ||
| Line 13: | Line 13: | ||
|$\mu(t)=0$ | |$\mu(t)=0$ | ||
|- | |- | ||
| − | |[[ | + | |[[Delta derivative | $\Delta$-derivative]] |
| − | |$f^{\Delta}(t)= | + | |$$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$$ |
|- | |- | ||
| − | |[[ | + | |[[Nabla derivative | $\nabla$-derivative]] |
| − | | $\ | + | |$$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$$ |
|- | |- | ||
| − | |[[ | + | |[[Delta integral | $\Delta$-integral]] |
| + | | $$\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$$ | ||
| + | |- | ||
| + | |[[Nabla derivative | $\nabla$-derivative]] | ||
| + | |$$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ | ||
| + | |- | ||
| + | |[[Delta exponential | $\Delta$-exponential]] | ||
| $\begin{array}{ll} | | $\begin{array}{ll} | ||
e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ | e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ | ||
| Line 25: | Line 31: | ||
&= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) | &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) | ||
\end{array}$ | \end{array}$ | ||
| + | |- | ||
| + | |[[Nabla exponential | $\nabla$-exponential]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Trig functions | $\mathrm{sin}_p(t,s)$]] | ||
| + | |- | ||
| + | |$\mathrm{\sin}_1(t,0)$ | ||
| + | | | ||
| + | |- | ||
| + | |$\mathrm{\cos}_p(t,0)$ | ||
| + | | | ||
| + | |- | ||
| + | |$\mathrm{\cos}_1(t,0)$ | ||
| + | | | ||
| + | |- | ||
|} | |} | ||
Revision as of 21:21, 20 October 2014
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.
| Generic element $t \in \mathbb{T}$: | $t=t$ |
| Jump operator: | $\sigma(t)=t$ |
| Graininess operator: | $\mu(t)=0$ |
| $\Delta$-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 | $$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$ |
| $\Delta$-exponential | $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$ |
| $\nabla$-exponential | |
| $\mathrm{sin}_p(t,s)$ | |
| $\mathrm{\sin}_1(t,0)$ | |
| $\mathrm{\cos}_p(t,0)$ | |
| $\mathrm{\cos}_1(t,0)$ |
