The following table summarizes common Laplace transforms that can be used to solve different Laplace transform problems:
| Function f(t) | Function F(s)=\mathscr{L}\{f(t)\} | Laplace transformation Proof |
|---|---|---|
| 1 | \dfrac{1}{s} | Check proof! |
| t^{n} | \dfrac{n!}{s^{n+1}},\: n\in\mathbb{Z}_{+} | Check proof! |
| \sin(bt) | \dfrac{b}{s^{2}+b^{2}} | Check proof! |
| \cos(bt) | \dfrac{s}{s^{2}+b^{2}} | Check proof! |
| e^{at} | \dfrac{1}{s-a} | Check proof! |
| t^{n}e^{at} | \dfrac{n!}{(s-a)^{n+1}},\quad n\in\mathbb{Z}_{+} | Check proof! |
| e^{at}\sin(bt) | \dfrac{b}{(s-a)^{2}+b^{2}} | Check proof! |
| e^{at}\cos(bt) | \dfrac{s-a}{(s-a)^{2}+b^{2}} | Check proof! |
| f(t-a)u(t-a) | e^{-as}F(s) | Check proof! |
| e^{at}f(t) | F(s-a) | Check proof! |
| f'(t) | sF(s)-f(0) | Check proof! |
| f''(t) | s^{2}F(s)-sf'(0)-f(0) | Check proof! |
| f^{(n)}(t) | s^{n}F(s)-s^{(n-1)}f(0)-\cdots-f^{(n-1)}(0) | Check proof! |
| t^{n}f(t) | (-1)^{n}\dfrac{d^{n}}{ds^{n}}F(s) | |
| (f\ast g)(t) | (-1)^{n}\dfrac{d^{n}}{ds^{n}}F(s) | Check proof! |
| \displaystyle\int_{0}^{t}f(\tau)d\tau | \dfrac{1}{s}F(s) | Check proof! |
| f(at) | \dfrac{1}{a}F\left(\dfrac{s}{a}\right) | Check proof! |
| f periodic with period T | \dfrac{1}{1-e^{-sT}}\displaystyle\int_{0}^{T}e^{-st}f(t)dt | Check proof! |
Download PDF version of common Laplace transforms:
