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 | ||
$$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$ | $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$ | ||
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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 /> | |
| − | + | [[Mozyraska-Torres logarithm is negative on (0,1)]]<br /> | |
| − | + | [[Mozyrska-Torres logarithm is positive on (1,infinity)]]<br /> | |
| + | [[Mozyrska-Torres logarithm composed with forward jump]]<br /> | ||
| + | [[Euler-Cauchy logarithm]]<br /> | ||
| + | [[Mozyrska-Torres logarithm tends to infinity]]<br /> | ||
=Special cases= | =Special cases= | ||
Latest revision as of 18:56, 11 December 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
Mozyrska-Torres logarithm tends to infinity
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)
