Difference between revisions of "Mozyrska-Torres logarithm"

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Let $\mathbb{T}$ be a time scale. For $t \in \mathbb{T} \cap (0,\infty)$, define
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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
 
$$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$
 
$$L_{\mathbb{T}}(t) = \displaystyle\int_1^t \dfrac{1}{\tau} \Delta \tau.$$
  
 
=Properties=
 
=Properties=
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*$L^{\Delta}_{\mathbb{T}}(t) = \dfrac{1}{t}$
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[[Delta derivative of Mozyrska-Torres logarithm]]<br />
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*$L_{\mathbb{T}}(1)=0$
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[[Mozyrska-Torres logarithm at 1]]<br />
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*$L_{\mathbb{R}}(t)=\log(t)$
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[[Mozyrska-Torres logarithm is increasing]]<br />
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*$L_{\mathbb{T}}(\cdot)$ is increasing and continuous
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[[Mozyraska-Torres logarithm is negative on (0,1)]]<br />
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*$L_{\mathbb{T}}(\sigma(t))=L_{\mathbb{T}}(t)+\mu(t)L_{\mathbb{T}}^{\Delta}(t)=L_{\mathbb{T}}(t)+\dfrac{\mu(t)}{t}$
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[[Mozyrska-Torres logarithm is positive on (1,infinity)]]<br />
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[[Mozyrska-Torres logarithm composed with forward jump]]<br />
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[[Euler-Cauchy logarithm]]<br />
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[[Mozyrska-Torres logarithm tends to infinity]]<br />
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=Special cases=
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[[Mozyrska-Torres logarithm on the reals]]<br />
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=See also=
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[[Bohner logarithm]]<br />
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[[Euler-Cauchy logarithm]]<br />
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[[Jackson logarithm]]<br />
  
 
=References=
 
=References=
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[http://arxiv.org/pdf/0809.4555.pdf Mozyrska, Dorota ; Torres, Delfim F. M. The natural logarithm on time scales. J. Dyn. Syst. Geom. Theor. 7 (2009), no. 1, 41--48.]
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{{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres|next=Delta derivative of Mozyrska-Torres logarithm}}
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[[Category:SpecialFunction]]

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)

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