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Cauchy index

释义

Definition
Examples
External links

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.

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 [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

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).

External links

The Cauchy Index

1 : Mathematical analysis

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