4.1.7. Table of Selected Laplace Transforms

These are from the Wikipedia page on the Laplace transform

Laplace Transform Properties
Function Time Domain \(s\)-Domain ROC
Unit pulse \(\delta(t)\) \(1\) \(\mathbb C\)
Delayed pulse \(\delta(t-\tau)\) \(e^{-\tau s}\) \(Re(s)>0\)
Unit step \(u(t)\) \(\frac{1}{s}\) \(Re(s)>0\)
Ramp \(t u(t)\) \(\frac{1}{s^2}\) \(Re(s)>0\)
Exponential decay \(e^{-\alpha t} u(t)\) \(\frac{1}{s+\alpha}\) \(Re(s)>-\alpha\)
Sine \(\sin(\omega t) u(t)\) \(\frac{\omega}{s^2+\omega^2}\) \(Re(s)>0\)
Cosine \(\cos(\omega t) u(t)\) \(\frac{s}{s^2+\omega^2}\) \(Re(s)>0\)
Exponentially Decaying Sine \(e^{-at}\sin(\omega t) u(t)\) \(\frac{\omega}{(s+a)^2+\omega^2}\) \(Re(s)>0\)
Exponentially Decaying Cosine \(e^{-at}\cos(\omega t) u(t)\) \(\frac{s+a}{(s+a)^2+\omega^2}\) \(Re(s)>0\)