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 | + | 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" | ||
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|- | |- | ||
|[[Delta hk|$h_k(t,s)$]] | |[[Delta hk|$h_k(t,s)$]] | ||
| − | | | + | |$h_k(t,s)=\dfrac{(t-s)^k}{k!}$ |
|[[Derivation of delta hk for T=R|derivation]] | |[[Derivation of delta hk for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla hk|$\hat{h}_k(t,s)$]] | |[[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]] | |[[Derivation of nabla hk for T=R|derivation]] | ||
|- | |- | ||
|[[Delta gk|$g_k(t,s)$]] | |[[Delta gk|$g_k(t,s)$]] | ||
| − | | | + | |$g_k(t,s)=\dfrac{(t-s)^k}{k!}$ |
|[[Derivation of delta gk for T=R|derivation]] | |[[Derivation of delta gk for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla gk|$\hat{g}_k(t,s)$]] | |[[Nabla gk|$\hat{g}_k(t,s)$]] | ||
| − | | | + | |$\hat{g}_k(t,s)=\dfrac{(t-s)^k}{k!}$ |
|[[Derivation of nabla gk for T=R|derivation]] | |[[Derivation of nabla gk for T=R|derivation]] | ||
|- | |- | ||
| − | |[[Delta exponential | $e_p(t,s) | + | |[[Delta exponential | $e_p(t,s)$]] |
| − | |$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$e_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
|[[Derivation of delta exponential T=R|derivation]] | |[[Derivation of delta exponential T=R|derivation]] | ||
|- | |- | ||
| − | |[[Nabla exponential | $\hat{e}_p(t,s) | + | |[[Nabla exponential | $\hat{e}_p(t,s)$]] |
| − | |$\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
|[[Derivation of nabla exponential T=R|derivation]] | |[[Derivation of nabla exponential T=R|derivation]] | ||
|- | |- | ||
| − | [[Gaussian bell]] | + | |[[Gaussian bell]] |
| − | | | + | |$\mathbf{E}(t)=e^{-\frac{t^2}{2}}$ |
|[[Derivation of Gaussian bell for T=R|derivation]] | |[[Derivation of Gaussian bell for T=R|derivation]] | ||
|- | |- | ||
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | |[[Delta sine | $\mathrm{sin}_p(t,s)=$]] | ||
| − | |$\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$\sin_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
|[[Derivation of delta sin sub p for T=R|derivation]] | |[[Derivation of delta sin sub p for T=R|derivation]] | ||
|- | |- | ||
| − | |$\mathrm{\sin}_1(t, | + | |$\mathrm{\sin}_1(t,s)$ |
| − | |$\sin(t)$ | + | |$\sin_1(t,s)=\sin(t-s)$ |
|[[Derivation of delta sin sub 1 for T=R|derivation]] | |[[Derivation of delta sin sub 1 for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]] | |[[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]] | |[[Derivation of nabla sine sub p for T=R|derivation]] | ||
|- | |- | ||
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]] | |[[Delta cosine|$\mathrm{\cos}_p(t,s)$]] | ||
| − | |$\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ | + | |$\cos_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
|[[Derivation of delta cos sub p for T=R|derivation]] | |[[Derivation of delta cos sub p for T=R|derivation]] | ||
|- | |- | ||
| − | |$\mathrm{\cos}_1(t, | + | |$\mathrm{\cos}_1(t,s)$ |
| − | |$\cos(t)$ | + | |$\cos_1(t,s)=\cos(t-s)$ |
|[[Derivation of delta cos sub 1 for T=R|derivation]] | |[[Derivation of delta cos sub 1 for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]] | |[[Nabla cosine|$\widehat{\cos}_p(t,s)$]] | ||
| − | | | + | |$\widehat{\cos}_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ |
|[[Derivation of nabla cos sub 1 for T=R|derivation]] | |[[Derivation of nabla cos sub 1 for T=R|derivation]] | ||
|- | |- | ||
|[[Delta sinh|$\sinh_p(t,s)$]] | |[[Delta sinh|$\sinh_p(t,s)$]] | ||
| − | | | + | |$\sinh_p(t,s)=$ |
|[[Derivation of delta sinh sub p for T=R|derivation]] | |[[Derivation of delta sinh sub p for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]] | |[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]] | ||
| − | | | + | |$\widehat{\sinh}_p(t,s)=$ |
|[[Derivation of nabla sinh sub p for T=R|derivation]] | |[[Derivation of nabla sinh sub p for T=R|derivation]] | ||
|- | |- | ||
|[[Delta cosh|$\cosh_p(t,s)$]] | |[[Delta cosh|$\cosh_p(t,s)$]] | ||
| − | | | + | |$\cosh_p(t,s)=$ |
|[[Derivation of delta cosh sub p for T=R|derivation]] | |[[Derivation of delta cosh sub p for T=R|derivation]] | ||
|- | |- | ||
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]] | |[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]] | ||
| − | | | + | |$\widehat{\cos}_p(t,s)=$ |
|[[Derivation of nabla cosh sub p for T=R|derivation]] | |[[Derivation of nabla cosh sub p for T=R|derivation]] | ||
|- | |- | ||
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|- | |- | ||
|[[Euler-Cauchy logarithm]] | |[[Euler-Cauchy logarithm]] | ||
| − | | | + | |$L(t,s)=$ |
|[[Derivation of Euler-Cauchy logarithm for T=R|derivation]] | |[[Derivation of Euler-Cauchy logarithm for T=R|derivation]] | ||
|- | |- | ||
|[[Bohner logarithm]] | |[[Bohner logarithm]] | ||
| − | | | + | |$L_p(t,s)=\log\left(\dfrac{p(t)}{p(s)}\right)$ |
|[[Derivation of the Bohner logarithm for T=R|derivation]] | |[[Derivation of the Bohner logarithm for T=R|derivation]] | ||
|- | |- | ||
|[[Jackson logarithm]] | |[[Jackson logarithm]] | ||
| − | | | + | |$\log_{\mathbb{T}} g(t)=$ |
|[[Derivation of the Jackson logarithm for T=R|derivation]] | |[[Derivation of the Jackson logarithm for T=R|derivation]] | ||
|- | |- | ||
|[[Mozyrska-Torres logarithm]] | |[[Mozyrska-Torres logarithm]] | ||
| − | | | + | |$L_{\mathbb{T}}(t)=$ |
|[[Derivation of the Mozyrska-Torres logarithm for T=R|derivation]] | |[[Derivation of the Mozyrska-Torres logarithm for T=R|derivation]] | ||
|- | |- | ||
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|- | |- | ||
|} | |} | ||
| + | |||
| + | =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.
| 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
- Robert J. Marks II, Ian A. Gravagne and John M. Davis: A generalized Fourier transform and convolution on time scales (2008)... (next): Section 2.1(a)
- Billy Jackson: Partial dynamic equations on time scales (2006)... (previous)... (next): Appendix
