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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__NOTOC__
$$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$
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Let $\mathbb{T}$ be a [[time_scale | time scale]], let $s \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regressive_function | regressive function]]. We define the trigonometric function $\sin_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by
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$$\sin_p(t,s;\mathbb{T})=\dfrac{e_{ip}(t,s;\mathbb{T})-e_{-ip}(t,s;\mathbb{T})}{2i}$$
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<div align="center">
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<gallery>
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File:Integerdeltasine,a=0.6,s=0plot.png | Plot of $\sin_{0.6}(t,0;\mathbb{Z})$.
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</gallery>
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</div>
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=Properties=
 
=Properties=
{{:Derivative of Delta sine}}
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[[Derivative of Delta sine]]<br />
{{:Sum of squares of delta cosine and delta sine}}
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[[Sum of squares of delta cosine and delta sine]]<br />
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[[Derivative of delta cosine]]<br />
  
=Relation to other special functions=
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=Examples=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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{{:Table:Time scale delta sine functions}}
<strong>Proposition:</strong> $\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0)$
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<div class="mw-collapsible-content">
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<center>{{:Delta special functions footer}}</center>
<strong>Proof:</strong> Compute
 
$$\begin{array}{ll}
 
\cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} (e_{ip}(t,t_0) + e_{-ip}(t,t_0) \\
 
&= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\
 
&= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\
 
&= -\sin_p(t,t_0)
 
\end{array}$$
 
as was to be shown. █
 
</div>
 
</div>
 
  
=Examples=
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[[Category:specialfunction]]
{| class="wikitable"
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[[Category:Definition]]
|+Time Scale Sine Functions
 
|-
 
|$\mathbb{T}$
 
|
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|$\sin_p(t,s)=  $
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|$\sin_p(t,s) = \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
| $\sin_p(t,s) = $
 
|-
 
| [[Square_integers | $\mathbb{Z}^2$]]
 
| $\sin_p(t,s) = $
 
|-
 
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
 
| $\sin_p(t,s) = $
 
|-
 
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
 
| $\sin_p(t,s) =$
 
|-
 
|[[Harmonic_numbers | $\mathbb{H}$]]
 
|$\sin_p(t,s) = $
 
|}
 

Latest revision as of 14:13, 28 January 2023

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


Properties

Derivative of Delta sine
Sum of squares of delta cosine and delta sine
Derivative of delta cosine

Examples

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

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$