If I derive the equation for a bandpass filter (in this case, critically damped) from the RLC transfer function, I get a different result than I would get otherwise by combining the transfer functions of a highpass and a lowpass filter.
$$H(s)=\frac{\frac{R}{L}s}{s^2+\frac{R}{L}s+\frac{1}{LC}}=\frac{2 \alpha s}{s^2+2 \alpha s+\omega_0^2}=\frac{2 \omega_c s}{s^2+2 \omega_c s+\omega_c^2}$$
Versus:
$$H(s)=\left(\frac{s}{s+\omega_c}\right)\left(\frac{\omega_c}{s+\omega_c}\right)=\frac{\omega_c s}{(s+\omega_c)^2}=\frac{\omega_c s}{s^2+2 \omega_c s+\omega_c^2}$$
As you can see, the two formulae differ by a factor of two. Which one is correct/why do they not agree?