Difference between revisions of "Nabla cosh"
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(Created page with "$$\hat{\cosh}_p(t,s)=\dfrac{\hat{e}_p(t,s)+\hat{e}_{-p}(t,s)}{2}$$ =Properties= <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</str...") |
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| − | $$\ | + | $$\widehat{\cosh}_p(t,s)=\dfrac{\widehat{e}_p(t,s)+\widehat{e}_{-p}(t,s)}{2}$$ |
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| − | <strong>Theorem:</strong> If $\alpha > 0$ with $\alpha^2\nu \in \mathcal{\nu}$, a [[regressive function]], then $\ | + | <strong>Theorem:</strong> If $\alpha > 0$ with $\alpha^2\nu \in \mathcal{\nu}$, a [[regressive function]], then $\widehat{\cosh}_{\gamma}(\cdot,s)$ and $\widehat{\sinh}_{\gamma}(\cdot,s)$ solve the $\nabla$ dynamic equation |
$$y^{\nabla \nabla}-\gamma^2 y=0.$$ | $$y^{\nabla \nabla}-\gamma^2 y=0.$$ | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
| − | #$\ | + | #$\widehat{\cosh}_p^{\nabla}(t,s)=p(t)\widehat{\sinh}_p(t,s)$, where $\widehat{\sinh}$ is the [[Nabla sinh|$\nabla$-$\sinh$]] function. |
| − | #$\ | + | #$\widehat{\cosh}^2_p(t,s)-\widehat{\sinh}^2_p(t,s)=\widehat{e}_{\nu p^2}(t,s)$ |
| − | #$\hat{\cosh}_p(t,s)-\ | + | #$\widehat{\cosh}_p(t,s) + \widehat{\sinh}_p(t,s)=\hat{e}_p(t,s)$ |
| + | #$\widehat{\cosh}_p(t,s)-\widehat{\sinh}_p(t,s)=\widehat{e}_{-p}(t,s)$ | ||
=References= | =References= | ||
[http://faculty.cord.edu/andersod/p20.pdf Nabla dynamic equations] | [http://faculty.cord.edu/andersod/p20.pdf Nabla dynamic equations] | ||
Revision as of 04:30, 6 March 2015
$$\widehat{\cosh}_p(t,s)=\dfrac{\widehat{e}_p(t,s)+\widehat{e}_{-p}(t,s)}{2}$$
Properties
Theorem: If $\alpha > 0$ with $\alpha^2\nu \in \mathcal{\nu}$, a regressive function, then $\widehat{\cosh}_{\gamma}(\cdot,s)$ and $\widehat{\sinh}_{\gamma}(\cdot,s)$ solve the $\nabla$ dynamic equation $$y^{\nabla \nabla}-\gamma^2 y=0.$$
Proof: █
- $\widehat{\cosh}_p^{\nabla}(t,s)=p(t)\widehat{\sinh}_p(t,s)$, where $\widehat{\sinh}$ is the $\nabla$-$\sinh$ function.
- $\widehat{\cosh}^2_p(t,s)-\widehat{\sinh}^2_p(t,s)=\widehat{e}_{\nu p^2}(t,s)$
- $\widehat{\cosh}_p(t,s) + \widehat{\sinh}_p(t,s)=\hat{e}_p(t,s)$
- $\widehat{\cosh}_p(t,s)-\widehat{\sinh}_p(t,s)=\widehat{e}_{-p}(t,s)$
