Laplace Transform

Standard Form:

$F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0}^{\infty }e^{-st}f(t)\,dt$

Sample Functions:

$F(s)={\mathcal {L}}\left\{1\right\}=\int _{0}^{\infty }e^{-st}\,dt=$ ${\frac {1}{s}}$

$F(s)={\mathcal {L}}\left\{t^{n}\right\}=\int _{0}^{\infty }e^{-st}t^{n}\,dt=$ ${\frac {n!}{s^{n+1}}}$

$F(s)={\mathcal {L}}\left\{e^{at}\right\}=\int _{0}^{\infty }e^{-st}e^{at}\,dt=$ ${\frac {1}{s-a}}$

$F(s)={\mathcal {L}}\left\{sin(\omega t)\right\}=\int _{0}^{\infty }e^{-st}sin(\omega t)\,dt=$ ${\frac {\omega }{s^{2}+\omega ^{2}}}$

$F(s)={\mathcal {L}}\left\{cos(\omega t)\right\}=\int _{0}^{\infty }e^{-st}cos(\omega t)\,dt=$ ${\frac {s}{s^{2}+\omega ^{2}}}$

$F(s)={\mathcal {L}}\left\{t^{n}g(t)\right\}=\int _{0}^{\infty }e^{-st}t^{n}g(t)\,dt=$ ${\frac {(-1)^{n}d^{n}G(s)}{ds^{n}}}{\mbox{ for}}~n\ {\mbox{= 1,2,...}}$

$F(s)={\mathcal {L}}\left\{tsin(\omega t)\right\}=\int _{0}^{\infty }e^{-st}tsin(\omega t)\,dt=$ ${\frac {2\omega s}{(s^{2}+\omega ^{2})^{2}}}$

$F(s)={\mathcal {L}}\left\{tcos(\omega t)\right\}=\int _{0}^{\infty }e^{-st}tcos(\omega t)\,dt=$ ${\frac {s^{2}-\omega ^{2}}{(s^{2}+\omega ^{2})^{2}}}$

$F(s)={\mathcal {L}}\left\{g(t)\right\}=\int _{0}^{\infty }e^{-st}g(t)\,dt=$ ${\frac {1}{a}}G\left({\frac {s}{a}}\right)$

$F(s)={\mathcal {L}}\left\{e^{at}g(t)\right\}=\int _{0}^{\infty }e^{-st}e^{at}g(t)\,dt=G(s-a)$

$F(s)={\mathcal {L}}\left\{e^{at}t^{n}\right\}=\int _{0}^{\infty }e^{-st}e^{at}t^{n}\,dt=$ ${\frac {n!}{(s-a)^{n+1}}}{\mbox{ for}}~n\ {\mbox{= 1,2,...}}$