Common Laplace Transform Pairs#
Transforms of Elementary Signals#
|
\(f(t)\) |
\(F(s)\) |
ROC |
|---|---|---|---|
1 |
\(\displaystyle \delta(t)\) |
\(\displaystyle 1\) |
All \(s\) |
2 |
\(\displaystyle \delta(t-a)\) |
\(\displaystyle e^{-as}\) |
All \(s\) |
3 |
\(\displaystyle u_0(t)\) |
\(\displaystyle \frac{1}{s}\) |
Re(\(s\)) > 0 |
4 |
\(\displaystyle -u_0(-t)\) |
\(\displaystyle \frac{1}{s}\) |
Re(\(s\)) < 0 |
5 |
\(\displaystyle t u_0(t)\) |
\(\displaystyle \frac{1}{s^2}\) |
Re(\(s\)) > 0 |
6 |
\(\displaystyle t^n u_0(t)\) |
\(\displaystyle \frac{n!}{s^{n+1}}\) |
Re(\(s\)) > 0 |
7 |
\(\displaystyle e^{-at}u_0(t)\) |
\(\displaystyle \frac{1}{s+a}\) |
Re(\(s\)) > \(-\)Re(\(a\)) |
8 |
\(\displaystyle -e^{-at}u_0(-t)\) |
\(\displaystyle \frac{1}{s+a}\) |
Re(\(s\))< \(-\)Re(\(a\)) |
9 |
\(\displaystyle t^n e^{-at} u_0(t)\) |
\(\displaystyle \frac{n!}{(s+a)^{n+1}}\) |
Re(\(s\)) > \(-\)Re(\(a\)) |
10 |
\(\displaystyle -t^n e^{-at} u_0(-t)\) |
\(\displaystyle \frac{n!}{(s+a)^{n+1}}\) |
Re(\(s\)) < \(-\)Re(\(a\)) |
11 |
\(\displaystyle \sin (\omega t) u_0(t)\) |
\(\displaystyle \frac{\omega}{s^2 + \omega^2}\) |
Re(\(s\)) > 0 |
12 |
\(\displaystyle \cos (\omega t) u_0(t)\) |
\(\displaystyle \frac{s}{s^2 + \omega^2}\) |
Re(\(s\)) > 0 |
13 |
\(\displaystyle e^{-at} \sin (\omega t) u_0(t)\) |
\(\displaystyle \frac{\omega}{(s + a)^2 + \omega^2}\) |
Re(\(s\)) > \(-\)Re(\(a\)) |
14 |
\(\displaystyle e^{-at}\cos (\omega t) u_0(t)\) |
\(\displaystyle \frac{s+a}{(s+a)^2 + \omega^2}\) |
Re(\(s\)) > \(-\)Re(\(a\)) |
See also: Wikibooks: Engineering_Tables/Laplace_Transform_Table and Laplace Transform—WolframMathworld for more complete references.