Difference between revisions of "Semigroup property of delta exponential"
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==Theorem== | ==Theorem== | ||
| − | Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds: | + | Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a [[forward regressive function]]. The following formula holds for all $s,t \in \mathbb{T}$: |
$$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ | $$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ | ||
where $e_p$ denotes the [[delta exponential]]. | where $e_p$ denotes the [[delta exponential]]. | ||
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==References== | ==References== | ||
| + | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta exponential|next=Delta exponential dynamic equation}}: Lemma $2.31$ | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:21, 8 February 2017
Theorem
Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a forward regressive function. The following formula holds for all $s,t \in \mathbb{T}$: $$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential.
Proof
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Lemma $2.31$
