Unilateral Laplace transform
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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 $$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t.$$
