Unilateral Laplace transform
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 $s$.
| 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
- The Laplace transform is linear, i.e. for constants $\alpha, \beta$ and Laplace-transformable functions $f,g$,
$$\mathscr{L}\{\alpha f + \beta g\} = \alpha \mathscr{L}\{f\} + \beta \mathscr{L}\{g\}.$$
- Assume there exist $M,\alpha > 0$ with
$$|a_k| \leq M \alpha_k$$ for all $k=0,1,2,\ldots$. Then for all $z$ where it exists, $$\mathscr{L}\left\{ \displaystyle\sum_{k=0}^{\infty} a_k h_k(\cdot,s) \right\}(z;s) = \displaystyle\sum_{k=0}^{\infty} a_k \mathscr{L}\{h_k(\cdot,s)\}(z;s) = \displaystyle\sum_{k=0}^{\infty} \dfrac{a_k}{z^{k+1}}.$$
- Let $m_z(t,s):=\displaystyle\int_s^t \dfrac{\Delta \tau}{1+\mu(\tau)z}$. Then
$$\dfrac{d}{dz} \mathscr{L}\{f\}(z;s) = -\mathscr{L}\{m_z(\sigma(\cdot),s)f\}(z;s).$$
