Delta sine

From timescalewiki
Jump to: navigation, search

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 $$\sin_p(t,s;\mathbb{T})=\dfrac{e_{ip}(t,s;\mathbb{T})-e_{-ip}(t,s;\mathbb{T})}{2i}$$


Derivative of Delta sine
Sum of squares of delta cosine and delta sine
Derivative of delta cosine


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

$\Delta$-special functions on time scales