Confusion about Laplace and frequency response of a electrical circuit
$$I(s) = \frac{V_m \omega_0 }{(s^2+\omega_0^2) \big(R + \frac{1}{s C}  + s L \big)} \tag{A}$$

For a linear system with sinusoidal voltage input, the current also will be sinusoidal. In which case, \$I(s)\$ can be written as,

$$I(s) = \frac{I_m \omega }{(s^2+\omega_0^2)}$$

Substituting this into \$(A)\$ and replacing \$s\$ with \$j\omega\$ should will result in \$(B)\$.