Laplace Transform

$$\mathscr{L}\{f(t)\}=F(s)$$$$\mathscr{L}\{f(t)\}=\int_0^\infty e^{-st} f(t) \, dt, \ \ {\rm for} \ s>0.$$$$s=\sigma+j\omega$$$$\omega = frequency$$

Given

$$KVL: V_R(j\omega)+V_L(j\omega)+V_C(j\omega)=V_{BarryWhite}(j\omega)$$$$Resistor: Z_R=R$$$$Inductor: Z_L(j\omega)=j\omega L$$$$Capacitor: Z_C(j\omega)=\frac1{j\omega C}$$

Substitution

$$I*Z_R+I*Z_L+I*Z_C=V_{BarryWhite}$$

Rearrange and solve for I

$$I=V_{BarryWhite}\frac1{Z_R+Z_L+Z_C}$$

Solve for V_C

$$V_C(j\omega)=V_{BarryWhite}(j\omega) \frac{Z_C}{Z_R+Z_L+Z_C}$$

substitute Z's and simplify

$$V_C(j\omega)=V_{BarryWhite}(j\omega) \frac{1}{1+j\omega RC + \omega^2 LC}$$