Difference between revisions of "Mozyrska-Torres logarithm"
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(Created page with "Let $\mathbb{T}$ be a time scale. For $t \in \mathbb{T} \cap (0,\infty)$, define $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$ =Properties= *$L^{...") |
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| + | [http://arxiv.org/pdf/0809.4555.pdf Mozyrska, Dorota ; Torres, Delfim F. M. The natural logarithm on time scales. J. Dyn. Syst. Geom. Theor. 7 (2009), no. 1, 41--48.] | ||
Revision as of 00:23, 22 May 2015
Let $\mathbb{T}$ be a time scale. For $t \in \mathbb{T} \cap (0,\infty)$, define $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$
Properties
- $L^{\Delta}_{\mathbb{T}}(t) = \dfrac{1}{t}$
- $L_{\mathbb{T}}(1)=0$
- $L_{\mathbb{R}}(t)=\log(t)$
- $L_{\mathbb{T}}(\cdot)$ is increasing and continuous
- $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
