# Difference between revisions of "Mozyrska-Torres logarithm"

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− | {{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres | + | {{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres|next=Delta derivative of Mozyrska-Torres logarithm}} |

[[Category:SpecialFunction]] | [[Category:SpecialFunction]] |

## Revision as of 15:29, 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

- $L_{\mathbb{T}}(\cdot)$ is increasing and continuous
- $L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$

# 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)