▁Properties ▁▁ ▁For ▁a ▁polynomial ▁to ▁be ▁Hur witz , ▁it ▁is ▁necessary ▁but ▁not ▁sufficient ▁that ▁all ▁of ▁its ▁coefficients ▁be ▁positive ▁( except ▁for ▁second - deg ree ▁polynomials , ▁which ▁also ▁doesn ' t ▁imply ▁su ffic iency ). ▁ ▁A ▁necessary ▁and ▁sufficient ▁condition ▁that ▁a ▁polynomial ▁is ▁Hur witz ▁is ▁that ▁it ▁passes ▁the ▁R outh – H ur witz ▁stability ▁criter ion . ▁ ▁A ▁given ▁polynomial ▁can ▁be ▁efficiently ▁tested ▁to ▁be ▁Hur witz ▁or ▁not ▁by ▁using ▁the ▁R outh ▁continued ▁fraction ▁expansion ▁technique . ▁ ▁The ▁properties ▁of ▁Hur witz ▁polynomials ▁are : ▁▁ ▁All ▁the ▁pol es ▁and ▁zeros ▁are ▁in ▁the ▁left ▁half ▁plane ▁or ▁on ▁its ▁boundary , ▁the ▁imag inary ▁axis . ▁ ▁Any ▁pol es ▁and ▁zeros ▁on ▁the ▁imag inary ▁axis ▁are ▁simple ▁( have ▁a ▁multip licity ▁of ▁one ). ▁ ▁Any ▁pol es ▁on ▁the ▁imag inary ▁axis ▁have ▁real ▁strictly ▁positive ▁resid ues , ▁and ▁similarly ▁at ▁any ▁zeros ▁on ▁the ▁imag inary ▁axis , ▁the ▁function ▁has ▁a ▁real ▁strictly ▁positive ▁derivative . ▁ ▁Over ▁the ▁right ▁half ▁plane , ▁the ▁minimum ▁value ▁of ▁the ▁real ▁part ▁of ▁a ▁PR ▁function ▁occurs ▁on ▁the ▁imag inary ▁axis ▁( because ▁the ▁real ▁part ▁of ▁an ▁analyt ic ▁function ▁constit utes ▁a ▁harm onic ▁function ▁over ▁the ▁plane , ▁and ▁therefore ▁satisfies ▁the ▁maximum ▁principle ). ▁ ▁The ▁polynomial ▁should ▁not ▁have ▁missing ▁powers ▁of ▁s . ▁ ▁References ▁▁▁ ▁Wayne ▁H . ▁Chen ▁( 1