Unilateral Laplace transform

From timescalewiki
Revision as of 18:55, 26 May 2014 by Tom (talk | contribs)
Jump to: navigation, search

Let $\mathbb{T}$ be a time scale and let $s \in \mathbb{T}$. If $f \in C_{rd}(\mathbb{T},\mathbb{C})$ ( rd-continuous) then we define the Laplace transform of $f$ about $s$ by the formula $$\mathscr{L}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t,$$ where $z$ lives in a domain $D \subset \mathbb{C}$ for which the integral converges. 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 $s$.

Laplace Transformations
Function $f(t;s)$ Laplace Transformation $\mathscr{L}\{f(\cdot;s)\}(z)$
$e_{\alpha}(t;s)$ $\dfrac{1}{z-\alpha}$
$h_n(t;s)$ $\dfrac{1}{z^{n+1}}$
$\sinh_{\alpha}(t;s)$ $\dfrac{\alpha}{z^2-\alpha^2}$
$\cosh_{\alpha}(t;s)$ $\dfrac{z}{z^2-\alpha^2}$
$\sin_{\alpha}(t;s)$ $\dfrac{\alpha}{z^2+\alpha^2}$
$\cos_{\alpha}(t;s)$ $\dfrac{z}{z^2+\alpha^2}$

Properties of Laplace Transforms