Delta sine
From timescalewiki
Let $\mathbb{T}$ be a time scale, let $s \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,s;\mathbb{T})=\dfrac{e_{ip}(t,s;\mathbb{T})-e_{-ip}(t,s;\mathbb{T})}{2i}$$
Plot of $\sin_{0.6}(t,0;\mathbb{Z})$.
Properties
Derivative of Delta sine
Sum of squares of delta cosine and delta sine
Derivative of delta cosine
Examples
| $\mathbb{T}$ | $\sin$$_p(t,s)= $ |
| $\mathbb{R}$ | |
| $\mathbb{Z}$ | $\dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$ |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
