Difference between revisions of "Mozyrska-Torres logarithm"

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[[Delta derivative of Mozyrska-Torres logarithm]]<br />
 
[[Delta derivative of Mozyrska-Torres logarithm]]<br />
 
[[Mozyrska-Torres logarithm at 1]]<br />
 
[[Mozyrska-Torres logarithm at 1]]<br />
*$L_{\mathbb{R}}(t)=\log(t)$
+
 
 
*$L_{\mathbb{T}}(\cdot)$ is increasing and continuous
 
*$L_{\mathbb{T}}(\cdot)$ is increasing and continuous
 
*$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
 
*$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
 +
 +
=Special cases=
 +
[[Mozyrska-Torres logarithm on the reals]]<br />
  
 
=See also=
 
=See also=

Revision as of 15:13, 21 October 2017

Let $\mathbb{T}$ be a time scale of positive numbers including $1$ and at least one other point $t$ such that $0< t < 1$. 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
Mozyrska-Torres logarithm at 1

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

Special cases

Mozyrska-Torres logarithm on the reals

See also

Bohner logarithm
Euler-Cauchy logarithm
Jackson logarithm

References

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