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 $$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$

Properties[edit]

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[edit]

Bohner logarithm
Euler-Cauchy logarithm
Jackson logarithm

References[edit]

Dorota Mozyrska and Delfim F. M. Torres: The Natural Logarithm on Time Scales (2009)... (previous)... (next): page 1