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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
- 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 (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 for a and b going to infinity).
Examples
- 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 . 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
- ^ "The Cauchy Index". deslab.mit.edu. Retrieved 2024-01-20.
External links
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