Difference between revisions of "Real numbers"

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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].
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The set $\mathbb{R}$ of real numbers is a [[time scale]]. In this time scale, all derivatives reduce to the classical [https://en.wikipedia.org/wiki/Derivative derivative] and the integrals reduce to the classical [https://en.wikipedia.org/wiki/Integral integral].
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|+$\mathbb{T}=\mathbb{R}$
 
|+$\mathbb{T}=\mathbb{R}$
−
|-
 
−
|Generic element $t \in \mathbb{T}$:
 
−
|$t=t$
 
 
|-
 
|-
 
|[[Forward jump]]:
 
|[[Forward jump]]:
 
|$\sigma(t)=t$
 
|$\sigma(t)=t$
 +
|[[Derivation of forward jump for T=R|derivation]]
 
|-
 
|-
−
|[[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|$h_k(t,s)$]]
 +
|$h_k(t,s)=\dfrac{(t-s)^k}{k!}$
 +
|[[Derivation of delta hk for T=R|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=\dfrac{(t-s)^k}{k!}$
 +
|[[Derivation of nabla hk for T=R|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=\dfrac{(t-s)^k}{k!}$
 +
|[[Derivation of delta gk for T=R|derivation]]
 
|-
 
|-
−
|[[Delta exponential | $\Delta$-exponential]]
+
|[[Nabla gk|$\hat{g}_k(t,s)$]]
−
| $\begin{array}{ll}
+
|$\hat{g}_k(t,s)=\dfrac{(t-s)^k}{k!}$
−
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) \\
+
|[[Derivation of nabla gk for T=R|derivation]]
−
&\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 | $\nabla$-exponential]]
+
|[[Delta exponential | $e_p(t,s)$]]
 +
|$e_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of delta exponential T=R|derivation]]
 +
|-
 +
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
 
|$\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 
|$\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of nabla exponential T=R|derivation]]
 +
|-
 +
|[[Gaussian bell]]
 +
|$\mathbf{E}(t)=e^{-\frac{t^2}{2}}$
 +
|[[Derivation of Gaussian bell for T=R|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
 +
|$\sin_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of delta sin sub p for T=R|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=\sin(t-s)$
 +
|[[Derivation of delta sin sub 1 for T=R|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of nabla sine sub p for T=R|derivation]]
 +
|-
 +
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
 +
|$\cos_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of delta cos sub p for T=R|derivation]]
 
|-
 
|-
−
|[[Trig functions | $\mathrm{sin}_p(t,s)$]]
+
|$\mathrm{\cos}_1(t,s)$
 +
|$\cos_1(t,s)=\cos(t-s)$
 +
|[[Derivation of delta cos sub 1 for T=R|derivation]]
 
|-
 
|-
−
|$\mathrm{\sin}_1(t,0)$
+
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
−
|$\sin(t)$
+
|$\widehat{\cos}_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
 +
|[[Derivation of nabla cos sub 1 for T=R|derivation]]
 
|-
 
|-
−
|$\mathrm{\cos}_p(t,s)$
+
|[[Delta sinh|$\sinh_p(t,s)$]]
−
|$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
+
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=R|derivation]]
 
|-
 
|-
−
|$\mathrm{\cos}_1(t,0)$
+
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
−
|$\cos(t)$
+
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=R|derivation]]
 
|-
 
|-
−
|[[Hilger circle]]  
+
|[[Delta cosh|$\cosh_p(t,s)$]]
−
|
+
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=R|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=R|derivation]]
 +
|-
 +
|[[Gamma function]]
 +
|$\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$
 +
|[[Derivation of gamma function for T=R|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=R|derivation]]
 +
|-
 +
|[[Bohner logarithm]]
 +
|$L_p(t,s)=\log\left(\dfrac{p(t)}{p(s)}\right)$
 +
|[[Derivation of the Bohner logarithm for T=R|derivation]]
 +
|-
 +
|[[Jackson logarithm]]
 +
|$\log_{\mathbb{T}} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=R|derivation]]
 +
|-
 +
|[[Mozyrska-Torres logarithm]]
 +
|$L_{\mathbb{T}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm 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]]
+
|[[Hilger circle]]  
 
|
 
|
 +
|[[Derivation of Hilger circle for T=R|derivation]]
 
|-
 
|-
 
|}
 
|}
 +
 +
=References=
 +
*{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=|next=Multiples of integers}}: Section 2.1(a)
 +
* {{PaperReference|Partial dynamic equations on time scales|2006|Billy Jackson||prev=Time scale|next=Quantum q greater than 1}}: Appendix
 +
 +
<center>{{:Time scales footer}}</center>

Latest revision as of 15:55, 15 January 2023

The set $\mathbb{R}$ of real numbers is a time scale. In this time scale, all derivatives reduce to the classical derivative and the integrals reduce to the classical integral.

$\mathbb{T}=\mathbb{R}$
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
$h_k(t,s)$ $h_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$g_k(t,s)$ $g_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$e_p(t,s)$ $e_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
Gaussian bell $\mathbf{E}(t)=e^{-\frac{t^2}{2}}$ derivation
$\mathrm{sin}_p(t,s)=$ $\sin_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=\sin(t-s)$ derivation
$\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=\cos(t-s)$ derivation
$\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\sinh_p(t,s)$ $\sinh_p(t,s)=$ derivation
$\widehat{\sinh}_p(t,s)$ $\widehat{\sinh}_p(t,s)=$ derivation
$\cosh_p(t,s)$ $\cosh_p(t,s)=$ derivation
$\widehat{\cosh}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ derivation
Gamma function $\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=\log\left(\dfrac{p(t)}{p(s)}\right)$ derivation
Jackson logarithm $\log_{\mathbb{T}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{T}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ derivation
Hilger circle derivation

References

Examples of time scales
$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set
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