Difference between revisions of "Delta cosine"
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| − | Let $\mathbb{T}$ be a [[time_scale | time scale]] | + | __NOTOC__ |
| + | Let $\mathbb{T}$ be a [[time_scale | time scale]], let $t_0 \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regressive_function | regressive function]]. We define the trigonometric functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$ | ||
$$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$ | $$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$ | ||
where $i=\sqrt{-1}$. | where $i=\sqrt{-1}$. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Integerdeltacosine,a=0.6,s=0plot.png | Plot of $\cos_{0.6}(t,0;\mathbb{Z})$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of delta cosine]]<br /> | |
| − | + | [[Sum of squares of delta cosine and delta sine]]<br /> | |
| + | [[Derivative of Delta sine]]<br /> | ||
| + | |||
| + | =Examples= | ||
| + | <center>{{:Table:Time scale delta cosine functions}}</center> | ||
| − | = | + | =See Also= |
| − | + | [[Delta sine]] <br /> | |
| + | [[Delta cosh]]<br /> | ||
| − | + | <center>{{:Delta special functions footer}}</center> | |
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| − | + | [[Category:specialfunction]] | |
| + | [[Category:Definition]] | ||
Latest revision as of 14:13, 28 January 2023
Let $\mathbb{T}$ be a time scale, let $t_0 \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$ $$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$ where $i=\sqrt{-1}$.
Plot of $\cos_{0.6}(t,0;\mathbb{Z})$.
Properties
Derivative of delta cosine
Sum of squares of delta cosine and delta sine
Derivative of Delta sine
Examples
| $\mathbb{T}$ | $\cos_p(t,s)= $ |
| $\mathbb{R}$ | |
| $\mathbb{Z}$ | |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
See Also
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
