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

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:

I_{s}r={\begin{cases}+1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=-\infty \;\land \;\lim _{x\downarrow s}r(x)=+\infty ,\\-1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=+\infty \;\land \;\lim _{x\downarrow s}r(x)=-\infty ,\\0,&{\text{otherwise.}}\end{cases}}

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 $I_s$ of r for each s located in the interval. We usually denote it by $I_{a}^{b}r$ .
We can then generalize to intervals of type $[-\infty,+\infty]$ 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

A rational function

Consider the rational function:

r(x)={\frac {4x^{3}-3x}{16x^{5}-20x^{3}+5x}}={\frac {p(x)}{q(x)}}.

We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5. Therefore r(x) has poles $x_{1}=0.9511$ , $x_{2}=0.5878$ , $x_{3}=0$ , $x_{4}=-0.5878$ and $x_{5}=-0.9511$ , i.e. $x_{j}=\cos((2i-1)\pi /2n)$ for $j=1,...,5$ . We can see on the picture that $I_{x_{1}}r=I_{x_{2}}r=1$ and $I_{x_{4}}r=I_{x_{5}}r=-1$ . For the pole in zero, we have $I_{x_{3}}r=0$ since the left and right limits are equal (which is because p(x) also has a root in zero). We conclude that $I_{-1}^{1}r=0=I_{-\infty }^{+\infty }r$ 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

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