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.[1]

Definition

edit
 
  • A generalization over the compact interval [a,b] is direct (when neither a nor b are poles of r(x)): 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 (by taking the limit of the Cauchy index over [a,b] for a and b going to infinity).

Examples

edit
 
A rational function
  • Consider the rational function:
 

We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5. Therefore, r(x) 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 (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).

References

edit
  1. ^ "The Cauchy Index". deslab.mit.edu. Retrieved 2024-01-20.
edit