Comment by periblepsis on How to prove the sinusoidal response to \$\sin\omega t\$ is the imaginary part of the response to \$e^{j \omega t}\$?

@LeonChang I almost think I know your question. But there is no way I'm going to try and unwind the Gordian knot output of a muddled mind to see if I can share your nightmares. So I'll just ask. Are you considering rationals, \$Y_s=\frac{N_s}{D_s}=\frac{a_2s^2+a_1s+a_0}{b_2s^2+b_1s+b_0‌​}\$, where \$a_i\$ and \$b_i\$ are real, positive constants such that \$a_i\ge 0\$ and \$b_i\gt 0\$? If so, you really should read Sallen & Key,: "A Practical Method of Designing RC Active Filters", 1954, TR-50 from MIT. And then just sit and re-arrange things long enough to consider annihilator operators.