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

• $L_{\mathbb{R}}(t)=\log(t)$
• $L_{\mathbb{T}}(\cdot)$ 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)