Laplace Transform Table
| $f(t)={\mathcal L}^{-1}\{F(s)\}$ | $F(s)={\mathcal L}\{f(t)\}$ | |
|---|---|---|
| $1$ | $\displaystyle\frac 1s$ | |
| $t^n$, $n$ a positive integer | $\displaystyle\frac{n!}{s^{n+1}}$ | |
| $t^a$, $a>-1$ | $\displaystyle\frac{\Gamma(a+1)}{s^{a+1}}$ | |
| $e^{at}$ | $\displaystyle\frac{1}{s-a}$ | |
| $t^ne^{at}$, $n$ a positive integer | $\displaystyle\frac{n!}{(s-a)^{n+1}}$ | |
| $\sin(at)$ | $\displaystyle\frac{a}{s^2+a^2}$ | |
| $\cos(at)$ | $\displaystyle\frac{s}{s^2+a^2}$ | |
| $t\sin(at)$ | $\displaystyle\frac{2as}{(s^2+a^2)^2}$ | |
| $t\cos(at)$ | $\displaystyle\frac{s^2-a^2}{(s^2+a^2)^2}$ | |
| $e^{at}\sin(bt)$ | $\displaystyle\frac{b}{(s-a)^2+b^2}$ | |
| $e^{at}\cos(bt)$ | $\displaystyle\frac{s-a}{(s-a)^2+b^2}$ | |
| $te^{at}\sin(bt)$ | $\displaystyle\frac{2(s-a)b}{((s-a)^2+b^2)^2}$ | |
| $te^{at}\cos(bt)$ | $\displaystyle\frac{(s-a)^2-b^2}{((s-a)^2+b^2)^2}$ | |
| $\delta(t-c)$ | $e^{-cs}$ | |
| $u(t-c)f(t-c)$ | $e^{-cs}{\mathcal L}\{f(t)\}$ | |
| $f'(t)$ | $s{\mathcal L}\{f(t)\}-f(0)$ | |
| $f^{(n)}(t)$ | $s^n{\mathcal L}\{f(t)\}-s^{n-1}f(0)-s^{n-2}f'(0) -\cdots-f^{(n-1)}(0)$ | |
| $\int_0^t f(T)\,dT$ | $\displaystyle\frac{1}{s} {\mathcal L}\{f(t)\}$ | |
| $\int_0^t f(t-T)g(T)\,dT$ | ${\mathcal L}\{f(t)\}{\mathcal L}\{g(t)\}$ |
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©2010, 2014 Andrew G. Bennett