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 located in the right half-plane and those located in the left half-plane. The complex polynomial f is such that
We must also assume that p has degree less than the degree of q.

Definition

  • The Cauchy index was first defined for a pole s of the rational function r by Augustin-Louis Cauchy in 1837 using one-sided limits as:
  • A generalization over the compact interval is direct : it is the sum of the Cauchy indices of r for each s located in the interval. We usually denote it by.
  • We can then generalize to intervals of type since the number of poles of r is a finite number.

    Examples

  • Consider the rational function:
We recognize in p and q respectively the Chebyshev polynomials of degree 3 and 5. Therefore, r has poles,,, and, i.e. for. We can see on the picture that and. For the pole in zero, we have since the left and right limits are equal.
We conclude that since q has only five roots, all in . We cannot use here the Routh–Hurwitz theorem as each complex polynomial with f = q + ip has a zero on the imaginary line.