MediaWiki API result

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            "6": {
                "pageid": 6,
                "ns": 0,
                "title": "Real numbers",
                "revisions": [
                    {
                        "contentformat": "text/x-wiki",
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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].\n\n{| class=\"wikitable\"\n|+$\\mathbb{T}=\\mathbb{R}$\n|-\n|[[Forward jump]]:\n|$\\sigma(t)=t$\n|[[Derivation of forward jump for T=R|derivation]]\n|-\n|[[Forward graininess]]:\n|$\\mu(t)=0$\n|[[Derivation of forward graininess for T=R|derivation]]\n|-\n|[[Backward jump]]:\n|$\\rho(t)=t$\n|[[Derivation of backward jump for T=R|derivation]]\n|-\n|[[Backward graininess]]:\n|$\\nu(t)=0$\n|[[Derivation of backward graininess for T=R|derivation]]\n|-\n|[[Delta derivative | $\\Delta$-derivative]]\n|$f^{\\Delta}(t)=\\displaystyle\\lim_{h\\rightarrow 0} \\dfrac{f(t+h)-f(t)}{h}=f'(t)$\n|[[Derivation of delta derivative for T=R|derivation]]\n|-\n|[[Nabla derivative | $\\nabla$-derivative]]\n|$f^{\\nabla}(t) =\\displaystyle\\lim_{h \\rightarrow 0} \\dfrac{f(t)-f(t-h)}{h}= f'(t)$\n|[[Derivation of nabla derivative for T=R|derivation]]\n|-\n|[[Delta integral | $\\Delta$-integral]]\n|$\\displaystyle\\int_s^t f(\\tau) \\Delta \\tau = \\int_s^t f(\\tau) d\\tau$\n|[[Derivation of delta integral for T=R|derivation]]\n|-\n|[[Nabla derivative | $\\nabla$-derivative]]\n|$\\displaystyle\\int_s^t f(\\tau) \\nabla \\tau = \\int_s^t f(\\tau) d\\tau$\n|[[Derivation of nabla integral for T=R|derivation]]\n|-\n|[[Delta hk|$h_k(t,s)$]]\n|$h_k(t,s)=\\dfrac{(t-s)^k}{k!}$\n|[[Derivation of delta hk for T=R|derivation]]\n|-\n|[[Nabla hk|$\\hat{h}_k(t,s)$]]\n|$\\hat{h}_k(t,s)=\\dfrac{(t-s)^k}{k!}$\n|[[Derivation of nabla hk for T=R|derivation]]\n|-\n|[[Delta gk|$g_k(t,s)$]]\n|$g_k(t,s)=\\dfrac{(t-s)^k}{k!}$\n|[[Derivation of delta gk for T=R|derivation]]\n|-\n|[[Nabla gk|$\\hat{g}_k(t,s)$]]\n|$\\hat{g}_k(t,s)=\\dfrac{(t-s)^k}{k!}$\n|[[Derivation of nabla gk for T=R|derivation]]\n|-\n|[[Delta exponential | $e_p(t,s)$]] \n|$e_p(t,s)=\\exp \\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of delta exponential T=R|derivation]]\n|-\n|[[Nabla exponential | $\\hat{e}_p(t,s)$]]\n|$\\hat{e}_p(t,s)=\\exp \\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of nabla exponential T=R|derivation]]\n|-\n|[[Gaussian bell]]\n|$\\mathbf{E}(t)=e^{-\\frac{t^2}{2}}$\n|[[Derivation of Gaussian bell for T=R|derivation]]\n|-\n|[[Delta sine | $\\mathrm{sin}_p(t,s)=$]]\n|$\\sin_p(t,s)=\\sin\\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of delta sin sub p for T=R|derivation]]\n|-\n|$\\mathrm{\\sin}_1(t,s)$\n|$\\sin_1(t,s)=\\sin(t-s)$\n|[[Derivation of delta sin sub 1 for T=R|derivation]]\n|-\n|[[Nabla sine|$\\widehat{\\sin}_p(t,s)$]]\n|$\\widehat{\\sin}_p(t,s)=\\sin\\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of nabla sine sub p for T=R|derivation]]\n|-\n|[[Delta cosine|$\\mathrm{\\cos}_p(t,s)$]]\n|$\\cos_p(t,s)=\\cos \\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of delta cos sub p for T=R|derivation]]\n|-\n|$\\mathrm{\\cos}_1(t,s)$\n|$\\cos_1(t,s)=\\cos(t-s)$\n|[[Derivation of delta cos sub 1 for T=R|derivation]]\n|-\n|[[Nabla cosine|$\\widehat{\\cos}_p(t,s)$]]\n|$\\widehat{\\cos}_p(t,s)=\\cos \\left( \\displaystyle\\int_s^t p(\\tau) d\\tau \\right)$\n|[[Derivation of nabla cos sub 1 for T=R|derivation]]\n|-\n|[[Delta sinh|$\\sinh_p(t,s)$]]\n|$\\sinh_p(t,s)=$\n|[[Derivation of delta sinh sub p for T=R|derivation]]\n|-\n|[[Nabla sinh|$\\widehat{\\sinh}_p(t,s)$]]\n|$\\widehat{\\sinh}_p(t,s)=$\n|[[Derivation of nabla sinh sub p for T=R|derivation]]\n|-\n|[[Delta cosh|$\\cosh_p(t,s)$]]\n|$\\cosh_p(t,s)=$\n|[[Derivation of delta cosh sub p for T=R|derivation]]\n|-\n|[[Nabla cosh|$\\widehat{\\cosh}_p(t,s)$]]\n|$\\widehat{\\cos}_p(t,s)=$\n|[[Derivation of nabla cosh sub p for T=R|derivation]]\n|-\n|[[Gamma function]]\n|$\\Gamma_{\\mathbb{R}}(x,s)=\\displaystyle\\int_0^{\\infty} \\left( \\dfrac{\\tau}{s} \\right)^{x-1}e^{-\\tau} d\\tau$\n|[[Derivation of gamma function for T=R|derivation]]\n|-\n|[[Euler-Cauchy logarithm]]\n|$L(t,s)=$\n|[[Derivation of Euler-Cauchy logarithm for T=R|derivation]]\n|-\n|[[Bohner logarithm]]\n|$L_p(t,s)=\\log\\left(\\dfrac{p(t)}{p(s)}\\right)$\n|[[Derivation of the Bohner logarithm for T=R|derivation]]\n|-\n|[[Jackson logarithm]]\n|$\\log_{\\mathbb{T}} g(t)=$\n|[[Derivation of the Jackson logarithm for T=R|derivation]]\n|-\n|[[Mozyrska-Torres logarithm]]\n|$L_{\\mathbb{T}}(t)=$\n|[[Derivation of the Mozyrska-Torres logarithm for T=R|derivation]]\n|-\n|[[Laplace transform]]\n|$\\mathscr{L}_{\\mathbb{R}}\\{f\\}(z;s)=\\displaystyle\\int_0^{\\infty} f(\\tau) e^{-z\\tau} d\\tau$\n|[[Derivation of Laplace transform for T=R|derivation]]\n|-\n|[[Hilger circle]] \n|\n|[[Derivation of Hilger circle for T=R|derivation]]\n|-\n|}\n\n=References=\n*{{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)\n* {{PaperReference|Partial dynamic equations on time scales|2006|Billy Jackson||prev=Time scale|next=Quantum q greater than 1}}: Appendix\n\n<center>{{:Time scales footer}}</center>"
                    }
                ]
            },
            "228": {
                "pageid": 228,
                "ns": 0,
                "title": "Reciprocal of delta exponential",
                "revisions": [
                    {
                        "contentformat": "text/x-wiki",
                        "contentmodel": "wikitext",
                        "*": "==Theorem==\nLet $\\mathbb{T}$ be a [[time scale]], let $t,s \\in \\mathbb{T}$, and let $p \\in \\mathcal{R}(\\mathbb{T},\\mathbb{C})$ be a [[regressive function]]. The following formula holds:\n$$\\dfrac{1}{e_p(t,s;\\mathbb{T})}=e_{\\ominus p}(s,t;\\mathbb{T}),$$\nwhere $e_p$ denotes the [[delta exponential]] and $\\ominus$ denotes [[circle minus]].\n\n==Proof==\n\n==References==\n\n[[Category:Theorem]]\n[[Category:Unproven]]"
                    }
                ]
            }
        }
    }
}