Difference between revisions of "Delta simple useful formula"
From timescalewiki
m (Tom moved page Simple useful formula to Delta simple useful formula) |
|||
| Line 1: | Line 1: | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| − | <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: | + | <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: █
