Driving Point Impedance

By inspection, we can write

$$R_d(s) = R + Ls \left\Vert \frac{1}{Cs}\right. = R +\frac{L/C}{Ls+\frac{1}{Cs}} = R + \frac{Ls}{1+LCs^2} = R + \frac{1}{C} \frac{s}{s^2+\frac{1}{LC}}.$$

where $\Vert$ denotes ``in parallel with,'' and we used the general formula, memorized by any electrical engineering student,

$$\zbox {R_1 \Vert R_2 = \frac{R_1 R_2}{R_1 + R_2}.}$$

That is, the impedance of the parallel combination of impedances $R_1$ and $R_2$ is given by the product divided by the sum of the impedances.