Difference between revisions of "Bilateral Laplace transform"
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$$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$ | $$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$ | ||
This integral is clearly a generalization of the [[Laplace transform]]. | This integral is clearly a generalization of the [[Laplace transform]]. | ||
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| + | =See also= | ||
| + | [[Laplace transform]]<br /> | ||
=References= | =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}}: $(3.1)$ | *{{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}}: $(3.1)$ | ||
*{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1 | *{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1 | ||
Revision as of 02:31, 16 January 2023
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.
See also
References
- John M. Davis, Ian A. Gravagne and Robert J. Marks II: Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series (2009)... (previous)... (next): $(3.1)$
- Tom Cuchta and Svetlin Georgiev: Analysis of the bilateral Laplace transform on time scales with applications (2021): Section 1
