Difference between revisions of "Unilateral Laplace transform of delta derivative"
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Revision as of 15:10, 21 January 2023
Theorem
If $\mathbb{T}$ is a time scale, 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 $f$.
Proof
Compute using integration by parts, $$\begin{array}{ll} \mathscr{L}\{f^{\Delta}\}(z) &= \displaystyle\int_0^{\infty} f^{\Delta}(\tau) e_{\ominus z}(\sigma(\tau),s) \Delta \tau \\ &= \end{array}$$ proving the claim. █
