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 |
\(e^{-at}\cos(\omega t) u(t)\) |
\(\frac{s+a}{(s+a)^2+\omega^2}\) |
\(Re(s)>0\) |