# What does it mean, for poles and zeroes to alternate; Foster's Theorem

I'm currently learning Foster's reactance theorem;

The most general driving-point impedance Z obtainable by means of a finite resistance-less network is a pure reactance which is odd rational function of frequency p and which is completely determined, except for a constant factor H, by assigning the resonant and anti-resonant frequencies subject to the condition that they alternate and include both zero and infinity. Any such impedance may be physically constructed wither by combining, in parallel, resonant circuits having impedances of the form iLp + 1/iCp, or by combining in series, anti-resonant circuits having impedances of the form [iCp+1/iLp]^-1

--A Reactance Theorem

Wikipedia comments on zeroes and poles;

A consequence of Foster's theorem is that the zeros and poles of any passive immittance function must alternate as frequency increases

My Question Is:

I don't understand, what they mean by "zeros and poles of any passive immittance function must alternate". Why does it "alternate", rather what is the meaning of "alternating"? Does this sentence in the theorem, "subject to the condition that they alternate and include both zero and infinity" convey the same result?

• It means as you go up in frequency you should encounter a pole then a zero then a pole etc. or vice versa. – user110971 Sep 22 '19 at 11:52