Before revision: The Cauchy index Given the system: Assuming no roots of lie on the imaginary axis, and letting = The number of roots of with negative real parts, and = The number of roots of with positive real parts then we have Expressing in polar form, we have where and from (2) note that where Now if the ith root of has a positive real part, then (using the notation y=(RE[y],IM[y])) and Similarly, if the ith root of has a negative real part, and Therefore, when the ith root of has a positive real part, and when the ith root of has a negative real part.
After revision: The Cauchy index Given the system: Assuming no roots of lie on the imaginary axis, and letting = The number of roots of with negative real parts, and = The number of roots of with positive real parts then we have Expressing in polar form, we have where and from (2) note that where Now if the ith root of has a positive real part, then (using the notation y=(RE[y],IM[y])) and Similarly, if the ith root of has a negative real part, and andandFrom (9) to (11) we find thatfrom (12) to (14) we find that when the ith root of has a positive real part, and when the ith root of has a negative real part.