Laplace Transform

Standard Form:

$F(s)={ℒ}\{f(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}f(t)dt$

Sample Functions:

$F(s)={ℒ}\left\{1\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}dt=$ $\frac{1}{s}$

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

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

$F(s)={ℒ}\{sin(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}sin(\omega t)dt=$ $\frac{\omega }{s^2+{\omega }^2}$

$F(s)={ℒ}\{cos(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}cos(\omega t)dt=$ $\frac{s}{s^2+{\omega }^2}$

$F(s)={ℒ}\{t^{n}g(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{t}^{n}g(t)dt=$ $$\begin{gathered} \frac{(-1{)}^n{d}^{n}G(s)}{d{s}^n}\text{ for}n \\ \text{= 1,2,...} \end{gathered}$$

$F(s)={ℒ}\{tsin(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}tsin(\omega t)dt=$ $\frac{2\omega s}{(s^2+{\omega }^2{)}^2}$

$F(s)={ℒ}\{tcos(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}tcos(\omega t)dt=$ $\frac{s^2-{\omega }^2}{(s^2+{\omega }^2{)}^2}$

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

$F(s)={ℒ}\{e^{at}g(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}g(t)dt=G(s-a)$

$F(s)={ℒ}\left\{e^{at}{t}^n\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}{t}^{n}dt=$ $$\begin{gathered} \frac{n!}{(s-a{)}^{n+1}}\text{ for}n \\ \text{= 1,2,...} \end{gathered}$$