Difference between revisions of "Mozyrska-Torres logarithm composed with forward jump"

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==Theorem==
 
==Theorem==
Let $\mathbb{T}$ be a [[time scale]]. Then,
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If $\mathbb{T}$ be a [[time scale]], then,
 
$$L_{\mathbb{T}}(\sigma(t)) = L_{\mathbb{T}}(t) + \dfrac{\mu(t)}{t},$$
 
$$L_{\mathbb{T}}(\sigma(t)) = L_{\mathbb{T}}(t) + \dfrac{\mu(t)}{t},$$
 
where $L_{\mathbb{T}}$ denotes the [[Mozyrska-Torres logarithm]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].
 
where $L_{\mathbb{T}}$ denotes the [[Mozyrska-Torres logarithm]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].
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==References==
 
==References==
 
{{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm is positive on (1,infinity)|next=Euler-Cauchy logarithm}}
 
{{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm is positive on (1,infinity)|next=Euler-Cauchy logarithm}}
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 15:13, 21 January 2023

Theorem

If $\mathbb{T}$ be a time scale, then, $$L_{\mathbb{T}}(\sigma(t)) = L_{\mathbb{T}}(t) + \dfrac{\mu(t)}{t},$$ where $L_{\mathbb{T}}$ denotes the Mozyrska-Torres logarithm, $\sigma$ denotes the forward jump, and $\mu$ denotes the forward graininess.

Proof

References

Dorota Mozyrska and Delfim F. M. Torres: The Natural Logarithm on Time Scales (2008)... (previous)... (next)