Difference between revisions of "Delta simple useful formula"
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| − | + | ==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: | |
$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$ | $$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$ | ||
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]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | * {{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) | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[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
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem 1.16 (iv)
