Difference between revisions of "Product of delta exponentials with fixed t and s"
From timescalewiki
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>[[Product of delta exponentials with fixed t and s|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and let $p,q \in \mathcal{R}\left(\mathbb{T},\mathbb{C}\right)$ be [[regressive function|regressive functions]]. The following formula holds: | <strong>[[Product of delta exponentials with fixed t and s|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and let $p,q \in \mathcal{R}\left(\mathbb{T},\mathbb{C}\right)$ be [[regressive function|regressive functions]]. The following formula holds: | ||
− | $$e_p(t,s)e_q(t,s)=e_{p \oplus q}(t,s),$$ | + | $$e_p(t,s;\mathbb{T})e_q(t,s;\mathbb{T})=e_{p \oplus q}(t,s;\mathbb{T}),$$ |
where $e_p$ denotes the [[delta exponential]] and $\oplus$ denotes [[circle plus]]. | where $e_p$ denotes the [[delta exponential]] and $\oplus$ denotes [[circle plus]]. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> |
Revision as of 23:20, 31 May 2016
Theorem: Let $\mathbb{T}$ be a time scale, $t,s \in \mathbb{T}$, and let $p,q \in \mathcal{R}\left(\mathbb{T},\mathbb{C}\right)$ be regressive functions. The following formula holds: $$e_p(t,s;\mathbb{T})e_q(t,s;\mathbb{T})=e_{p \oplus q}(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential and $\oplus$ denotes circle plus.
Proof: █