Difference between revisions of "Delta sine"

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Let $\mathbb{T}$ be a [[time_scale | time scale]] and let $t_0 \in \mathbb{T}$ and let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regressive_function | regressive function]]. We define the trigonometric function $\sin_p \colon \mathbb{T} \rightarrow \mathbb{R}$ by
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$$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$
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{| class="wikitable"
 
{| class="wikitable"
 
|+Time Scale Sine Functions
 
|+Time Scale Sine Functions

Revision as of 06:04, 1 March 2015

Let $\mathbb{T}$ be a time scale and let $t_0 \in \mathbb{T}$ and let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$

Time Scale Sine Functions
$\mathbb{T}$
$\mathbb{R}$ $\sin_p(t,s)= $
$\mathbb{Z}$ $\sin_p(t,s) = \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$
$h\mathbb{Z}$ $\sin_p(t,s) = $
$\mathbb{Z}^2$ $\sin_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q > 1$ $\sin_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q < 1$ $\sin_p(t,s) =$
$\mathbb{H}$ $\sin_p(t,s) = $