Difference between revisions of "Bilateral Laplace transform"
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Let $\mathbb{T}$ be a [[time scale]]. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by | Let $\mathbb{T}$ be a [[time scale]]. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by | ||
| − | $$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s).$$ | + | $$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$ |
This integral is clearly a generalization of the [[Laplace transform]]. | This integral is clearly a generalization of the [[Laplace transform]]. | ||
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| + | =References= | ||
| + | *{{PaperReference|Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series|2009|John M. Davis|author2=Ian A. Gravagne|author3=Robert J. Marks II|prev=findme|next=findme}}: $(3.1)$ | ||
Revision as of 17:34, 7 July 2017
Let $\mathbb{T}$ be a time scale. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by $$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$ This integral is clearly a generalization of the Laplace transform.
