Difference between revisions of "Delta cosh"
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| + | Let $p$ and $-\mu p^2$ be [[regressive function|regressive functions]]. Then the $\Delta$ hyperbolic cosine function is defined by | ||
| + | $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$ | ||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> Let $p\in C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then | ||
| + | $$\cosh^{\Delta}_p = p\sinh_p,$$ | ||
| + | where $\sinh_p$ is the [[Delta cosh | $\sin_p$]] function. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =Relation to other functions= | ||
| + | <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"> | ||
| + | <strong>Theorem:</strong> Let $\gamma$ be a nonzero regressive real number, then a general solution of the second order dynamic equation is | ||
| + | $$y^{\Delta \Delta}-\gamma^2 y= 0$$ | ||
| + | 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> | ||
| + | |||
| + | =Examples= | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Time Scale $\Delta$-$\cosh_1$ Functions | |+Time Scale $\Delta$-$\cosh_1$ Functions | ||
Revision as of 06:27, 1 March 2015
Let $p$ and $-\mu p^2$ be regressive functions. Then the $\Delta$ hyperbolic cosine function is defined by $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
Properties
Theorem: Let $p\in C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then $$\cosh^{\Delta}_p = p\sinh_p,$$ where $\sinh_p$ is the $\sin_p$ function.
Proof: █
Relation to other functions
Theorem: $\cosh^2_p - \sinh^2_p = e_{-\mu p^2}$
Proof: █
Theorem: Let $\gamma$ be a nonzero regressive real number, then a general solution of the second order dynamic equation is $$y^{\Delta \Delta}-\gamma^2 y= 0$$ is given by $$y(t) = c_1 \cosh_{\gamma}(t,s) + c_2 \sinh_{\gamma}(t,s).$$
Proof: █
Examples
| $\mathbb{T}=$ | $\cosh_1(t,0)=$ |
| $\mathbb{R}$ | $\cosh_1(t,0)=\cosh(t)$ |
| $\mathbb{Z}$ | |
| $h\mathbb{Z}$ | $\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) $ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
