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_sr = \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 $$I_a^br$$.
 * 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

 * 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 $$I_{-1}^1r=0=I_{-\infty}^{+\infty}r$$ since q(x) has only five roots, all in [&minus;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).