Unilateral Laplace transform of delta derivative

Theorem

If $\mathbb{T}$ is a time scale and $f \colon \mathbb{T} \rightarrow \mathbb{C}$, then $$\mathscr{L}_{\mathbb{T}}\{f^{\Delta}\}(z;s) = -f(s) + z\mathscr{L}\{f\}(z),$$ where $\mathscr{L}_{\mathbb{T}}$ denotes the unilateral Laplace transform and $f^{\Delta}$ denotes the delta derivative of $f$.