Difference between revisions of "Semigroup property of delta exponential"
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| − | <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 [[regressive]]. The following formula holds: | + | <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: |
$$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ | $$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ | ||
where $e_p$ denotes the [[delta exponential]]. | where $e_p$ denotes the [[delta exponential]]. | ||
Revision as of 22:49, 31 May 2016
Theorem: 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: $$e_p(t,r;\mathbb{T})e_p(r,s;\mathbb{T})=e_p(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential.
Proof: █
