Electrical Elements in Laplace Domain

Time	Laplace
$v=Ri$	$V=RI$
$v=L\frac{di}{dt}$	$V=LIs$
$v=\frac{1}{c}\int idt$	$V=\frac{1}{C}\frac{I}{s}$

Impedance is a generalization of resistance:

$$R=\frac{V(s)}{I(s)}=Z(s)$$

An inductor is:

$$Z(s)=\frac{V(s)}{I(s)}=\frac{LIs}{I}=Ls$$

A capacitor is:

$$Z(s)=\frac{V(s)}{I(s)}=\frac{\frac{1}{c}\frac{I}{s}}{I}=\frac{1}{Cs}$$

Parallel and series impedances work the same way that resistances do:

$$\begin{array}{rl}\text{Parallel: }& \frac{1}{Z_{eq}}=\frac{1}{Z_1}+\frac{1}{Z_2}\\ \text{Series: }& Z_{eq}=Z_1+Z_2\end{array}$$