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

Definition

Examples

We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5. Therefore, r(x) has poles x_1=0.9511, x_2=0.5878, x_3=0, x_4=-0.5878 and x_5=-0.9511, i.e. for j = 1,...,5. We can see on the picture that and. For the pole in zero, we have I_{x_3}r=0 since the left and right limits are equal (which is because p(x) also has a root in zero). We conclude that since q(x) has only five roots, all in [−1,1]. We cannot use here the Routh–Hurwitz theorem as each complex polynomial with f(iy) = q(y) + ip(y) has a zero on the imaginary line (namely at the origin).