# Signals and Systems/Table of Laplace Transforms

## Laplace Transform

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

## Inverse Laplace Transform

${\displaystyle {\mathcal {L}}^{-1}\left\{F(s)\right\}={1 \over {2\pi i}}\int _{c-i\infty }^{c+i\infty }e^{ft}F(s)\,ds=f(t)}$

## Laplace Transform Properties

Property Definition
Linearity ${\displaystyle {\mathcal {L}}\left\{af(t)+bg(t)\right\}=aF(s)+bG(s)}$
Differentiation ${\displaystyle {\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0^{-})}$

${\displaystyle {\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})}$
${\displaystyle {\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})}$

Frequency Division ${\displaystyle {\mathcal {L}}\{tf(t)\}=-F'(s)}$

${\displaystyle {\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)}$

Frequency Integration ${\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma }$
Time Integration ${\displaystyle {\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{u(t)*f(t)\right\}={1 \over s}F(s)}$
Scaling ${\displaystyle {\mathcal {L}}\left\{f(at)\right\}={1 \over a}F\left({s \over a}\right)}$
Initial value theorem ${\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}}$
Final value theorem ${\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}$
Frequency Shifts ${\displaystyle {\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)}$

${\displaystyle {\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)}$

Time Shifts ${\displaystyle {\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)}$

${\displaystyle {\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)}$

Convolution Theorem ${\displaystyle {\mathcal {L}}\{f(t)*g(t)\}=F(s)G(s)}$

Where:

${\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}$
${\displaystyle g(t)={\mathcal {L}}^{-1}\{G(s)\}}$
${\displaystyle s=\sigma +j\omega }$

## Table of Laplace Transforms

No. Time Domain
${\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}$
Laplace Domain
${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$
1 ${\displaystyle {\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds}$ ${\displaystyle \int _{-\infty }^{\infty }x(t)e^{-st}dt}$
2 ${\displaystyle \delta (t)\,}$ ${\displaystyle 1\,}$
3 ${\displaystyle \delta (t-a)\,}$ ${\displaystyle e^{-as}\,}$
4 ${\displaystyle u(t)\,}$ ${\displaystyle {\frac {1}{s}}}$
5 ${\displaystyle u(t-a)\,}$ ${\displaystyle {\frac {e^{-as}}{s}}}$
6 ${\displaystyle tu(t)\,}$ ${\displaystyle {\frac {1}{s^{2}}}}$
7 ${\displaystyle t^{n}u(t)\,}$ ${\displaystyle {\frac {n!}{s^{n+1}}}}$
8 ${\displaystyle {\frac {1}{\sqrt {\pi t}}}u(t)}$ ${\displaystyle {\frac {1}{\sqrt {s}}}}$
9 ${\displaystyle e^{at}u(t)\,}$ ${\displaystyle {\frac {1}{s-a}}}$
10 ${\displaystyle t^{n}e^{at}u(t)\,}$ ${\displaystyle {\frac {n!}{(s-a)^{n+1}}}}$
11 ${\displaystyle \cos(\omega t)u(t)\,}$ ${\displaystyle {\frac {s}{s^{2}+\omega ^{2}}}}$
12 ${\displaystyle \sin(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{s^{2}+\omega ^{2}}}}$
13 ${\displaystyle \cosh(\omega t)u(t)\,}$ ${\displaystyle {\frac {s}{s^{2}-\omega ^{2}}}}$
14 ${\displaystyle \sinh(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{s^{2}-\omega ^{2}}}}$
15 ${\displaystyle e^{at}\cos(\omega t)u(t)\,}$ ${\displaystyle {\frac {s-a}{(s-a)^{2}+\omega ^{2}}}}$
16 ${\displaystyle e^{at}\sin(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{(s-a)^{2}+\omega ^{2}}}}$
17 ${\displaystyle {\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)}$ ${\displaystyle {\frac {1}{(s^{2}+\omega ^{2})^{2}}}}$
18 ${\displaystyle {\frac {t}{2\omega }}\sin \omega t}$ ${\displaystyle {\frac {s}{(s^{2}+\omega ^{2})^{2}}}}$
19 ${\displaystyle {\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)}$ ${\displaystyle {\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}}$