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:

$I_{s}r=\{\begin{array}{ll}+1,& \text{if }\underset{x\uparrow s}{\lim }r(x)=-\mathrm{\infty }\wedge \underset{x\downarrow s}{\lim }r(x)=+\mathrm{\infty },\\ -1,& \text{if }\underset{x\uparrow s}{\lim }r(x)=+\mathrm{\infty }\wedge \underset{x\downarrow s}{\lim }r(x)=-\mathrm{\infty },\\ 0,& \text{otherwise.}\end{array}$

• 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 $[-\mathrm{\infty },+\mathrm{\infty }]$ 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 $I_{-1}^{1}r=0=I_{-\mathrm{\infty }}^{+\mathrm{\infty }}r$ 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