Difference between revisions of "Semigroup property of delta exponential"

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==Theorem==
<strong>[[Semigroup property of delta exponential|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and let $p \in \mathcal{R}\left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:
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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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<strong>Proof:</strong>  █
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==Proof==
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==References==
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta exponential|next=Delta exponential dynamic equation}}: Lemma $2.31$
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[[Category:Theorem]]
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[[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