Difference between revisions of "Mozyrska-Torres logarithm"
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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 | 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 | ||
$$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$ | $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$ | ||
Revision as of 15:33, 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)
