Signals and Systems/Table of Laplace Transforms

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Laplace Transform[edit]

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

Inverse Laplace Transform[edit]

 
\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[edit]

Property Definition
Linearity \mathcal{L}\left\{a f(t) + b g(t) \right\}  = a F(s)  + b G(s)
Differentiation \mathcal{L}\{f'\}  = s \mathcal{L}\{f\} - f(0^-)

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

Frequency Division \mathcal{L}\{ t f(t)\}  = -F'(s)

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

Frequency Integration \mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_s^\infty F(\sigma)\, d\sigma
Time Integration \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  \mathcal{L} \left\{ f(at) \right\} = {1 \over a} F \left ( {s \over a} \right )
Initial value theorem f(0^+)=\lim_{s\to \infty}{sF(s)}
Final value theorem f(\infty)=\lim_{s\to 0}{sF(s)}
Frequency Shifts \mathcal{L}\left\{ e^{at} f(t) \right\}  = F(s - a)

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

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

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

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

Where:

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

Table of Laplace Transforms[edit]

  Time Domain Laplace Domain
x(t) = \mathcal{L}^{-1}\left\{ X(s) \right\} X(s) = \mathcal{L} \left\{ x(t) \right\}
1  \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s)e^{st}ds  \int_{-\infty}^\infty x(t)e^{-st}dt
2  \delta (t) \,  1 \,
3  \delta (t-a)\,  e^{-as}\,
4  u(t) \,  \frac{1}{s}
5  u(t-a)\,  \frac{e^{-as}}{s}
6  t u(t) \,  \frac{1}{s^2}
7  t^nu(t) \,  \frac{n!}{s^{n+1}}
8  \frac{1}{\sqrt{\pi t}}u(t)  \frac{1}{\sqrt{s}}
9  e^{at}u(t) \,  \frac{1}{s-a}
10  t^n e^{at}u(t) \,  \frac{n!}{(s-a)^{n+1}}
11   \cos (\omega t) u(t) \,  \frac{s}{s^2+\omega^2}
12   \sin (\omega t) u(t) \,  \frac{\omega}{s^2+\omega^2}
13   \cosh (\omega t) u(t) \,  \frac{s}{s^2-\omega^2}
14  \sinh (\omega t) u(t) \,  \frac{\omega}{s^2-\omega^2}
15  e^{at}  \cos (\omega t) u(t) \,  \frac{s-a}{(s-a)^2+\omega^2}
16  e^{at} \sin (\omega t) u(t) \,  \frac{\omega}{(s-a)^2+\omega^2}
17  \frac{1}{2\omega^3}(\sin \omega t-\omega t \cos \omega t)  \frac{1}{(s^2+\omega^2)^2}
18  \frac{t}{2\omega} \sin \omega t  \frac{s}{(s^2+\omega^2)^2}
19  \frac{1}{2\omega}(\sin \omega t+\omega t \cos \omega t)  \frac{s^2}{(s^2+\omega^2)^2}