Difference between revisions of "Mozyrska-Torres logarithm composed with forward jump"
From timescalewiki
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==Theorem== | ==Theorem== | ||
| − | + | 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]] | ||
| + | [[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)
