Mozyrska-Torres logarithm

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)