Delta cosine
From timescalewiki
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 | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
