Difference between revisions of "Delta simple useful formula"

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<strong>[[Simple useful formula|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:
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<strong>[[Simple useful formula|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:
 
$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$
 
$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$
 
where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].
 
where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].

Revision as of 23:13, 31 May 2016

Theorem: Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a regressive function. The following formula holds: $$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential, $\sigma$ denotes the forward jump, and $\mu$ denotes the forward graininess.

Proof: