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 $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regressive_function | regressive function]]. We define the trigonometric functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$  
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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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=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 />
=Relation to other special functions=
 
{{:Derivative of Delta sine}}
 
  
 
=Examples=
 
=Examples=
{{:Table:Time scale delta cosine functions}}
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<center>{{:Table:Time scale delta cosine functions}}</center>
  
 
=See Also=
 
=See Also=
[[Delta cosine]] <br />
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[[Delta sine]] <br />
[[Delta hyperbolic cosine]]<br />
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[[Delta cosh]]<br />
  
 
<center>{{:Delta special functions footer}}</center>
 
<center>{{:Delta special functions footer}}</center>
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[[Category:specialfunction]]
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[[Category:Definition]]

Latest revision as of 14:13, 28 January 2023

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


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$