Difference between revisions of "Unilateral Laplace transform"
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Let $\mathbb{T}$ be a [[time_scale | 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 | Let $\mathbb{T}$ be a [[time_scale | 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 | ||
$$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t.$$ | $$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t.$$ | ||
| + | |||
| + | Let $s \in \mathbb{T}$. Let $\alpha$ be a non-negative [[regressive_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" | ||
| + | |+Laplace Transformations | ||
| + | |- | ||
| + | |Function | ||
| + | |Laplace Transformation | ||
| + | |- | ||
| + | |$e_{\alpha}(t;s)$ | ||
| + | |$\dfrac{1}{z-\alpha}$ | ||
| + | |- | ||
| + | |$h_n(t;s)$ | ||
| + | |$\dfrac{1}{z^{n+1}}$ | ||
| + | |- | ||
| + | |$\sinh_{\alpha}(t;s)$ | ||
| + | |$\dfrac{\alpha}{z^2-\alpha^2}$ | ||
| + | |- | ||
| + | |$\cosh_{\alpha}(t;s)$ | ||
| + | |$\dfrac{z}{z^2-\alpha^2}$ | ||
| + | |- | ||
| + | |$\sin_{\alpha}(t;s)$ | ||
| + | |$\dfrac{\alpha}{z^2+\alpha^2}$ | ||
| + | |- | ||
| + | |$\cos_{\alpha}(t;s)$ | ||
| + | |$\dfrac{z}{z^2+\alpha^2}$ | ||
| + | |- | ||
| + | |||
| + | |} | ||
Revision as of 18:44, 26 May 2014
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 |
| $e_{\alpha}(t;s)$ | $\dfrac{1}{z-\alpha}$ |
| $h_n(t;s)$ | $\dfrac{1}{z^{n+1}}$ |
| $\sinh_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2-\alpha^2}$ |
| $\cosh_{\alpha}(t;s)$ | $\dfrac{z}{z^2-\alpha^2}$ |
| $\sin_{\alpha}(t;s)$ | $\dfrac{\alpha}{z^2+\alpha^2}$ |
| $\cos_{\alpha}(t;s)$ | $\dfrac{z}{z^2+\alpha^2}$ |
