Difference between revisions of "Mozyrska-Torres logarithm"

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=Properties=
 
=Properties=
*$L^{\Delta}_{\mathbb{T}}(t) = \dfrac{1}{t}$
+
[[Delta derivative of Mozyrska-Torres logarithm]]<br />
 
*$L_{\mathbb{T}}(1)=0$
 
*$L_{\mathbb{T}}(1)=0$
 
*$L_{\mathbb{R}}(t)=\log(t)$
 
*$L_{\mathbb{R}}(t)=\log(t)$

Revision as of 20:54, 17 September 2016

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

Delta derivative of Mozyrska-Torres logarithm

  • $L_{\mathbb{T}}(1)=0$
  • $L_{\mathbb{R}}(t)=\log(t)$
  • $L_{\mathbb{T}}(\cdot)$ is increasing and continuous
  • $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$

References

Dorota Mozyrska and Delfim F. M. Torres: The Natural Logarithm on Time Scales (2009)... (previous)... (next): page 1