Difference between revisions of "Delta sine"

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(Properties)
(Relation to other special functions)
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=Relation to other special functions=
 
=Relation to other special functions=
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{{:Derivative of delta cosine}}
<strong>Proposition:</strong> $\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0)$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> Compute
 
$$\begin{array}{ll}
 
\cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} (e_{ip}(t,t_0) + e_{-ip}(t,t_0) \\
 
&= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\
 
&= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\
 
&= -\sin_p(t,t_0)
 
\end{array}$$
 
as was to be shown. █
 
</div>
 
</div>
 
  
 
=Examples=
 
=Examples=

Revision as of 17:58, 21 March 2015

Let $\mathbb{T}$ be a time scale and let $t_0 \in \mathbb{T}$ and let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$

Properties

Theorem

The following formula holds: $$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ where $\sin_p$ denotes the $\Delta$-$\sin_p$ function and $\cos_p$ denotes the $\Delta$-$\cos_p$ function.

Proof

Compute $$\begin{array}{ll} \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) \\ &= \dfrac{ip}{2i} ( e_{ip}(t,t_0) + e_{-ip}(t,t_0) ) \\ &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\ &= p\cos_p(t,t_0), \end{array}$$ as was to be shown. █

References

Theorem

The following formula holds: $$\cos_p^2(t,t_0)+\sin_p^2(t,t_0)=e_{\mu p^2}(t,t_0),$$ where $\cos_p$ denotes the $\Delta$-$\cos_p$ function and $\sin_p$ denotes the $\Delta$-$\sin_p$ function.

Proof

References

Relation to other special functions

Theorem

The following formula holds: $$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ where $\cos^{\Delta}_p$ denotes the delta derivative of the delta cosine function and $\sin_p$ denotes the delta sine function.

Proof

Compute $$\begin{array}{ll} \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\ &= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\sin_p(t,t_0), \end{array}$$ as was to be shown. █

References

Examples

Time Scale Sine Functions
$\mathbb{T}$
$\mathbb{R}$ $\sin_p(t,s)= $
$\mathbb{Z}$ $\sin_p(t,s) = \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$
$h\mathbb{Z}$ $\sin_p(t,s) = $
$\mathbb{Z}^2$ $\sin_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q > 1$ $\sin_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q < 1$ $\sin_p(t,s) =$
$\mathbb{H}$ $\sin_p(t,s) = $