Difference between revisions of "Delta simple useful formula"

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==Theorem==
<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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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]].
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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 derivative at right-dense|next=Delta derivative of sum}}: Theorem 1.16 (iv)
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 06:08, 10 June 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

References