Difference between revisions of "Delta cosine"
From timescalewiki
(→Properties) |
|||
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
__NOTOC__ | __NOTOC__ | ||
| − | Let $\mathbb{T}$ be a [[time_scale | time scale]] | + | 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 $i=\sqrt{-1}$. | ||
| Line 13: | Line 13: | ||
[[Derivative of delta cosine]]<br /> | [[Derivative of delta cosine]]<br /> | ||
[[Sum of squares of delta cosine and delta sine]]<br /> | [[Sum of squares of delta cosine and delta sine]]<br /> | ||
| − | [[Derivative of | + | [[Derivative of Delta sine]]<br /> |
=Examples= | =Examples= | ||
| − | {{:Table:Time scale delta cosine functions}} | + | <center>{{:Table:Time scale delta cosine functions}}</center> |
| + | |||
=See Also= | =See Also= | ||
[[Delta sine]] <br /> | [[Delta sine]] <br /> | ||
| Line 24: | Line 25: | ||
[[Category:specialfunction]] | [[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 | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
