Difference between revisions of "Delta cosh"

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Let $p \in C_{rd}$ and $-\mu p^2$ be a [[regressive function]]. Then the $\Delta$-hyperbolic cosine function is defined by
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$$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
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<div align="center">
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<gallery>
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File:Integercosh,a=0.6,s=0plot.png|Graph of $\cosh_{0.6}(t,0;\mathbb{Z})$.
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</gallery>
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</div>
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=Properties=
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[[Derivative of delta cosh]]<br />
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[[Delta cosh minus delta sinh]]<br />
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[[Delta hyperbolic trigonometric second order dynamic equation]]<br />
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=Examples=
 
{| class="wikitable"
 
{| class="wikitable"
 
|+Time Scale $\Delta$-$\cosh_1$ Functions
 
|+Time Scale $\Delta$-$\cosh_1$ Functions
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
| $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}} = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) \right)$
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| $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $
 
|-
 
|-
 
| [[Square_integers | $\mathbb{Z}^2$]]
 
| [[Square_integers | $\mathbb{Z}^2$]]
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|
 
|}
 
|}
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<center>{{:Delta special functions footer}}</center>
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[[Category:specialfunction]]

Revision as of 21:21, 9 June 2016

Let $p \in C_{rd}$ and $-\mu p^2$ be a regressive function. Then the $\Delta$-hyperbolic cosine function is defined by $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$

Properties

Derivative of delta cosh
Delta cosh minus delta sinh
Delta hyperbolic trigonometric second order dynamic equation

Examples

Time Scale $\Delta$-$\cosh_1$ Functions
$\mathbb{T}=$ $\cosh_1(t,0)=$
$\mathbb{R}$ $\cosh_1(t,0)=\cosh(t)$
$\mathbb{Z}$
$h\mathbb{Z}$ $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$