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