Difference between revisions of "Unilateral Laplace transform"
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| − | Let $\mathbb{T}$ be a [[time_scale | time scale]] and let $ | + | Let $\mathbb{T}$ be a [[time_scale | time scale]] and let $s \in \mathbb{T}$. If $f \in C_{rd}(\mathbb{T},\mathbb{C})$ ([[continuity | rd-continuous]]) then we define the Laplace transform of $f$ about $s$ by the formula |
| − | $$\mathscr{L}\{f\}(z) = \displaystyle\ | + | $$\mathscr{L}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t,$$ |
| − | + | where $z$ lives in a domain $D \subset \mathbb{C}$ for which the integral converges. Let $\alpha$ be a non-negative [[regressive_function | regressive]] constant larger than $s$. We use the notation "$f(t;s)$" to denote we are thinking of $f$ as a function of $t$ with parameter $s$. | |
| − | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Laplace Transformations | |+Laplace Transformations | ||
|- | |- | ||
| − | |Function | + | |Function $f(t;s)$ |
| − | |Laplace Transformation | + | |Laplace Transformation $\mathscr{L}\{f(\cdot;s)\}(z)$ |
|- | |- | ||
|$e_{\alpha}(t;s)$ | |$e_{\alpha}(t;s)$ | ||
| Line 28: | Line 27: | ||
|$\dfrac{z}{z^2+\alpha^2}$ | |$\dfrac{z}{z^2+\alpha^2}$ | ||
|- | |- | ||
| + | |} | ||
| − | + | ==Properties of Laplace Transforms== | |
Revision as of 18:55, 26 May 2014
Let $\mathbb{T}$ be a time scale and let $s \in \mathbb{T}$. If $f \in C_{rd}(\mathbb{T},\mathbb{C})$ ( rd-continuous) then we define the Laplace transform of $f$ about $s$ by the formula $$\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 $\mathscr{L}\{f(\cdot;s)\}(z)$ |
| $e_{\alpha}(t;s)$ | $\dfrac{1}{z-\alpha}$ |
| $h_n(t;s)$ | $\dfrac{1}{z^{n+1}}$ |
| $\sinh_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2-\alpha^2}$ |
| $\cosh_{\alpha}(t;s)$ | $\dfrac{z}{z^2-\alpha^2}$ |
| $\sin_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2+\alpha^2}$ |
| $\cos_{\alpha}(t;s)$ | $\dfrac{z}{z^2+\alpha^2}$ |
