4. Find the following. (1) \(L(\delta(t-2)*u(t-1))\) (2) \(L(\delta(t-1)*tu(t-3))\) (3) \(L(\sin(t-1)u(t-1)*\sinh 2t)\)
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Step 1: The Laplace transform of the product of two functions is given by the convolution theorem: $$L[f(t)g(t)] = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} F(s)G(s-z)dz$$ where $F(s)$ and $G(s)$ are the Laplace transforms of $f(t)$ and $g(t)$, respectively. Show more…
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