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 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.

[1] We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5.

Therefore, r(x) has poles

= 0.9511

,

x

= cos ⁡ ( ( 2 i − 1 ) π

We can see on the picture that

For the pole in zero, we have

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).