Delayed Inputs and Convolution — Cheat sheet

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Core rule

L[H(t−a)f(t−a)]=e−asF(s).\mathcal L[H(t-a)f(t-a)]=e^{-as}F(s).

Watch for

An ordinary step in forcing usually leaves the state continuous while changing its derivative. An impulse can cause a jump instead. Separate the forcing’s discontinuities from the response’s continuity requirements.