# Laplace Transform

$\mathscr L{f(t)} = F(s)$
$f(t)$ $F(s)$ Condition on s
a $\frac{a}{x}$ $s\ >\ 0$
$t^n,\ n = 0,1,2...$ $\frac{n!}{s^{n+1}}$ $s\ >\ 0$
$e^{at}$ $\frac{1}{s-a}$ $s\ >\ a$
$sin\ at$ $\frac{a}{s^2 + a^2}$ $s\ >\ 0$
$cos\ at$ $\frac{s}{s^2 + a^2}$ $s\ >\ 0$
$sinh\ at$ $\frac{a}{s^2 - a^2}$ $s\ >\ \mid a \mid$
$cosh\ at$ $\frac{s}{s^2 - a^2}$ $s\ >\ \mid a \mid$

## First Shifting Proferty

If $\mathscr L {f(t)} = F(s)$, then $\mathscr L(e^{at}f(t)) = F(s-a)$

## Multiplication by $t^n$ Property

If $\mathscr L {f(t)} = F(s)$, then $\mathscr L(t^n\ f(t)) = \frac{d^nF}{ds^n},\ n = 1,2,3...$

## Laplace Transform of Unit Step Functions, $\mathscr L{H(t-a)}$

 $\mathscr L(H(t-a)) = \frac{e^{-as}}{s},\ a\ >\ 0$

## Laplace Transform of $\mathscr L{f(t-a)H(t-a)}$

Second shifting property:

 $\mathscr L{f(t-a)H(t-a)} = e^{-as}\ \mathscr L {f(t)} = e^{-as}F(s)$

## Laplace Transform of Dirac Delta Function $\mathscr L(\delta(t-a))$

 $\mathscr L (\delta (t)) = 1$
 $\mathscr L (\delta (t-a)) = e^{-as},\ a\ >\ 0$
 $\mathscr L (f(t) \delta (t-a)) = e^{-as}f(a),\ a\ >\ 0$

# Inverse Laplace Transform

 $\mathscr L^{-1}(F(s-a)) = e^{at} f(t)$ $\mathscr L^{-1}\frac({e^{-as}}{s}) =H(t-a)$ $\mathscr L^{-1}(a^{-as}F(s)) = f(t-a)H(t-a)$ $\mathscr L^{-1}(e^{-as}) =\delta(t-a)$ $\mathscr L^{-1}(e^{-as}f(a)) =f(t)\delta(t-a)$

## Convolution Theorem

|—| |$\mathscr L F(s)G(s) = \int_{0}^{t} f(u)g(t-u)\ du$|