# Math Help - laplace transform

1. ## laplace transform

Use the formula for the derivative of a Laplace transform to show that if:

$F(s)= arctan(a/s)$

then:

$f(t)= (sin(at))/t$

2. use inverse Laplace transformation

$\displaystyle \frac {1}{2\pi i} \lim _{n\to \infty} \int _{\gama -iT} ^{\gama +iT} e^{st}F(S) \; ds$

3. One of the fundamental properties os the Laplace Trasform is that if $\mathcal{L} \{f(t)\} = F(s)$ then...

$\displaystyle \mathcal{L} \{\frac{f(t)}{t}\} = \int_{s}^{\infty} F(u)\ du$ (1)

... so that if $f(t)= \sin at$ then $\displaystyle \mathcal{L} \{f(t)\} = \frac{a}{s^{2} + a^{2}}$ so that is ...

$\displaystyle \mathcal {L} \{\frac{\sin a t}{t}\} = \int_{s}^{\infty} \frac{a\ du}{u^{2} + a^{2}} = \frac{\pi}{2} - \tan^{-1} \frac{s}{a} = \cot^{-1} \frac{s}{a}$ (2)

Kind regards

$\chi$ $\sigma$