Delta cosine

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 $i=\sqrt{-1}$.

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  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

Time Scale Cosine Functions

$\mathbb{T}$	$\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 sine
Delta cosh

$\Delta$-special functions on time scales
$\cos_p$	$\cosh_p$	$e_p$	$g_k$	$h_k$	$\sin_p$	$\sinh_p$