Cauchy Index

In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of

r(x) = p(x)/q(x)

over the real line is the difference between the number of roots of f(z) located in the right half-plane and those located in the left half-plane. The complex polynomial f(z) is such that

f(iy) = q(y) + ip(y).

We must also assume that p has degree less than the degree of q.