Difference between revisions of "Delta sinh"

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Let $p$ and $-\mu p^2$ be [[regressive function|regressive functions]]. Then the $\Delta$ hyperbolic sine function is defined by
 
Let $p$ and $-\mu p^2$ be [[regressive function|regressive functions]]. Then the $\Delta$ hyperbolic sine function is defined by
$$\sinh_p(t,s) = \dfrac{e_p(t,s)-e_{-p}(t,s)}{2i}.$$
+
$$\sinh_p(t,s) = \dfrac{e_p(t,s)-e_{-p}(t,s)}{2}.$$
 +
 
 +
<div align="center">
 +
<gallery>
 +
File:Integerdeltasinh,a=0.6,s=0plot.png|Graph of $\sinh_{0.6}(t,0;\mathbb{Z})$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
{{:Derivative of delta sinh}}
+
[[Derivative of delta sinh]]<br />
{{:Derivative of delta cosh}}
+
[[Derivative of delta cosh]]<br />
{{:Delta cosh minus delta sinh}}
+
[[Delta cosh minus delta sinh]]<br />
{{:Delta hyperbolic trigonometric second order dynamic equation}}
+
[[Delta hyperbolic trigonometric second order dynamic equation]]<br />
 
 
 
 
  
=Relation to other functions=
+
<center>{{:Delta special functions footer}}</center>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> $\cosh^2_p - \sinh^2_p = e_{-\mu p^2}$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
[[Category:specialfunction]]
<strong>Theorem:</strong> Let $\gamma$ be a nonzero regressive real number, then a general solution of the second order dynamic equation is
+
[[Category:Definition]]
$$y^{\Delta \Delta}-\gamma^2 y= 0$$
 
is given by
 
$$y(t) = c_1 \cosh_{\gamma}(t,s) + c_2 \sinh_{\gamma}(t,s).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 

Latest revision as of 14:13, 28 January 2023

Let $p$ and $-\mu p^2$ be regressive functions. Then the $\Delta$ hyperbolic sine function is defined by $$\sinh_p(t,s) = \dfrac{e_p(t,s)-e_{-p}(t,s)}{2}.$$

Properties

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

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$