4.1.7. Table of Selected Laplace Transforms¶
These are from the Wikipedia page on the Laplace transform
| 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 | \(e^{-at}\cos(\omega t) u(t)\) | \(\frac{s+a}{(s+a)^2+\omega^2}\) | \(Re(s)>0\) |