Transfer function request

Obviously for this one the gain can't keep rising forever but that's OK.

Ahh not sure, but isn't there something about having the poles on the left side of the plane... or it's an oscillator.

George H.

Actually I bungled my requirements anyway, what I need is the input impedance function to the network to look like that, not the transfer function.

In that case with passive components at least I think what I'm looking for is just a RCLC circuit in both cases, RC high-pass or low pass and then a series LC band-reject in parallel with the R or C, depending from the output to ground

this elegant passive structure should work for the second type I mentioned. at low frequency the input impedance is just R1 + R2. then it dips due to the complex-conjugate zeroes. Then rises again due to CC poles. Then dips out to high frequency due to the single pole.

Thanx to Wolfram Alpha/Mathematica for help with the analysis

You mentioned you meant this as a "driving point immittance." What you gave is not a rational function: s*(s^2 + 2*zeta*s + 1)

So, automatically, the answer is "no, it can't be synthesized (realized)."

For a driving point immittance to be realizable requires

1. F(s) is real if s is real.
2. Real(F(s)) >= 0 if Real(s) >= 0

These are the Brune conditions. (Otto Brune, 1931)

F(s) is the immittance function in s, whether impedance or admittance.

Some other interesting rules are derived if it is something like a lossless one-port or a doubly terminated ladder.

I should have said that it has to be a (positive real) rational function. 1 & 2 ensure the rational function is "positive real."

Yep. Infinite gain at infinite frequency is indeed a tall order