Things You Should Know -

Laplace Transform

$F(s)=\int_{0}^{\infty}f(t)e^{-st}dt$

$s = \sigma+i\omega$

Derivative Transforms

$f^{'}(t)\Rightarrow{s}F(s)-f(0^{-})$

$f^{''}(t)\Rightarrow{s^{2}}F(s)-sf(0^{-})-f^{'}(0^{-})$

Integral Transform

$\int_{0}^{t}f(\tau)d\tau\Rightarrow\dfrac{1}{s}F(s)$

Electrical Components

Resistor

$v=iR\Rightarrow{V(s)}=RI(s)$

Inductor

$v=L\dfrac{di}{dt}\Rightarrow{V(s)}=L[sI(s)-i(0^{-})]$

Capacitor

$i=C\dfrac{dv}{dt}\Rightarrow{I(s)}=C[sV(s)-v(0^{-})]$