Difference between revisions of "Delta cosine"

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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 functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$  
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__NOTOC__
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Let $\mathbb{T}$ be a [[time_scale | time scale]], let $t_0 \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regressive_function | regressive function]]. We define the trigonometric functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$  
 
$$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$
 
$$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$
 
where $i=\sqrt{-1}$.
 
where $i=\sqrt{-1}$.
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<div align="center">
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<gallery>
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File:Integerdeltacosine,a=0.6,s=0plot.png | Plot of $\cos_{0.6}(t,0;\mathbb{Z})$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
{{:Derivative of delta cosine}}
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[[Derivative of delta cosine]]<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 sine]]<br />
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=Examples=
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{{:Table:Time scale delta cosine functions}}
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=See Also=
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[[Delta sine]] <br />
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[[Delta cosh]]<br />
  
=Relation to other special functions=
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<center>{{:Delta special functions footer}}</center>
{{:Derivative of Delta sine}}
 
  
=Examples=
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[[Category:specialfunction]]
{| class="wikitable"
 
|+Time Scale Cosine Functions
 
|-
 
|$\mathbb{T}$
 
|
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|$\cos_p(t,s)=  $
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|$\cos_p(t,s) = $
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
| $\cos_p(t,s) = $
 
|-
 
| [[Square_integers | $\mathbb{Z}^2$]]
 
| $\cos_p(t,s) = $
 
|-
 
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
 
| $\cos_p(t,s) = $
 
|-
 
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
 
| $\cos_p(t,s) =$
 
|-
 
|[[Harmonic_numbers | $\mathbb{H}$]]
 
|$\cos_p(t,s) = $
 
|}
 

Revision as of 02:27, 11 June 2016

Let $\mathbb{T}$ be a time scale, let $t_0 \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$ $$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$ where $i=\sqrt{-1}$.

Properties

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

Examples

Time Scale Cosine Functions
$\mathbb{T}$ $\cos_p(t,s)= $
$\mathbb{R}$
$\mathbb{Z}$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

See Also

Delta sine
Delta cosh

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$