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 />
 
+
[[Mozyrska-Torres logarithm is increasing]]<br />
*$L_{\mathbb{T}}(\cdot)$ is increasing and continuous
+
[[Mozyraska-Torres logarithm is negative on (0,1)]]<br />
*$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
+
[[Mozyrska-Torres logarithm is positive on (1,infinity)]]<br />
 +
[[Mozyrska-Torres logarithm composed with forward jump]]<br />
 +
[[Euler-Cauchy logarithm]]<br />
  
 
=Special cases=
 
=Special cases=

Revision as of 15:32, 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
Mozyrska-Torres logarithm is increasing
Mozyraska-Torres logarithm is negative on (0,1)
Mozyrska-Torres logarithm is positive on (1,infinity)
Mozyrska-Torres logarithm composed with forward jump
Euler-Cauchy logarithm

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 (2008)... (next)