Difference between revisions of "Mozyrska-Torres logarithm"
From timescalewiki
| Line 8: | Line 8: | ||
*$L_{\mathbb{T}}(\cdot)$ is increasing and continuous | *$L_{\mathbb{T}}(\cdot)$ is increasing and continuous | ||
*$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$ | *$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$ | ||
| + | |||
| + | =See also= | ||
| + | [[Bohner logarithm]]<br /> | ||
| + | [[Euler-Cauchy logarithm]]<br /> | ||
| + | [[Jackson logarithm]]<br /> | ||
=References= | =References= | ||
Revision as of 17:09, 11 February 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
- $L_{\mathbb{R}}(t)=\log(t)$
- $L_{\mathbb{T}}(\cdot)$ is increasing and continuous
- $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
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): page 1
