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 $$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$ where $e_{\ominus z}$ denotes the delta exponential and $\ominus z$ denotes forward circle minus.

See also

Unilateral Laplace transform
Cuchta-Georgiev Fourier transform