Unilateral Laplace transform
From timescalewiki
Let $\mathbb{T}$ be a time scale and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. For an appropriate $D \subset \mathbb{R}$ we define the Laplace transform of $f$, $\mathscr{L}\{f\} \colon D \rightarrow \mathbb{R}$ by the formula $$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t.$$
Let $s \in \mathbb{T}$. Let $\alpha$ be a non-negative regressive constant larger than $s$. We use the notation "$f(t;s)$" to denote we are thinking of $f$ as a function of $t$ with parameter $s$.
| Function | Laplace Transformation |
| $e_{\alpha}(t;s)$ | $\dfrac{1}{z-\alpha}$ |
| $h_n(t;s)$ | $\dfrac{1}{z^{n+1}}$ |
| $\sinh_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2-\alpha^2}$ |
| $\cosh_{\alpha}(t;s)$ | $\dfrac{z}{z^2-\alpha^2}$ |
| $\sin_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2+\alpha^2}$ |
| $\cos_{\alpha}(t;s)$ | $\dfrac{z}{z^2+\alpha^2}$ |
