Difference between revisions of "Mozyrska-Torres logarithm"
From timescalewiki
(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^{...") |
(No difference)
|
Revision as of 00:21, 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
- $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
