Engineering Handbook/Mathematics/Laplace Transformation

Laplace Transform

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

Inverse Laplace Transform

${ℒ}^{-1}\{F(s)\}=\frac{1}{2\pi i}\underset{c-i\mathrm{\infty }}{\overset{c+i\mathrm{\infty }}{\int }}{e}^{ft}F(s)ds=f(t)$

Laplace Transform Properties

Property	Definition
Linearity	$ℒ\{af(t)+bg(t)\}=aF(s)+bG(s)$
Differentiation	$ℒ\{f^{\prime }\}=sℒ\{f\}-f(0^{-})$ $ℒ\{f^{″}\}=s^2ℒ\{f\}-sf(0^{-})-f^{\prime }(0^{-})$ $ℒ\left\{f^{(n)}\right\}=s^nℒ\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$
Frequency Division	$ℒ\{tf(t)\}=-F^{\prime }(s)$ $ℒ\{t^{n}f(t)\}=(-1{)}^n{F}^{(n)}(s)$
Frequency Integration	$ℒ\left\{\frac{f(t)}{t}\right\}=\underset{s}{\overset{\mathrm{\infty }}{\int }}F(\sigma )d\sigma$
Time Integration	$ℒ\{\underset{0}{\overset{t}{\int }}f(\tau )d\tau \}=ℒ\{u(t)*f(t)\}=\frac{1}{s}F(s)$
Scaling	$ℒ\{f(at)\}=\frac{1}{a}F\left(\frac{s}{a}\right)$
Initial value theorem	$f(0^{+})=\underset{s\to \mathrm{\infty }}{\lim }sF(s)$
Final value theorem	$f(\mathrm{\infty })=\underset{s\to 0}{\lim }sF(s)$
Frequency Shifts	$ℒ\{e^{at}f(t)\}=F(s-a)$ ${ℒ}^{-1}\{F(s-a)\}=e^{at}f(t)$
Time Shifts	$ℒ\{f(t-a)u(t-a)\}=e^{-as}F(s)$ ${ℒ}^{-1}\{e^{-as}F(s)\}=f(t-a)u(t-a)$
Convolution Theorem	$ℒ\{f(t)*g(t)\}=F(s)G(s)$

Where:

$f(t)={ℒ}^{-1}\{F(s)\}$

$g(t)={ℒ}^{-1}\{G(s)\}$

$s=\sigma +j\omega$

Table of Laplace Transforms

No.	Time Domain $x(t)={ℒ}^{-1}\{X(s)\}$	Laplace Domain $X(s)=ℒ\{x(t)\}$
1	$\frac{1}{2\pi j}\underset{\sigma -j\mathrm{\infty }}{\overset{\sigma +j\mathrm{\infty }}{\int }}X(s)e^{st}ds<math>|<math>\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-st}dt$
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	$tu(t)$	$\frac{1}{s^2}$
7	$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$
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	$\frac{1}{2\omega }(\sin \omega t+\omega t\cos \omega t)$	$\frac{s^2}{(s^2+{\omega }^2{)}^2}$