Cauchy index

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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. This interpretation suggests that for each polynomial g(z) such that

g(iy)=f(iy),

the repartition of the roots of g (over a half-plane) will be the same as that of f.

[edit] Definition

The Cauchy index was first defined for a pole s of the rational function r by Augustin Louis Cauchy in 1837 as:

$I_sr=\left\{\begin{matrix}+1 & \textrm{if } \lim_{x\to s,x<s}=-\infty, & \lim_{x\to s,x>s}=+\infty\\-1 & \textrm{if } \lim_{x\to s,x<s}=\infty, & \lim_{x\to s,x>s}=-\infty\\0 & \textrm{else.}&\end{matrix}\right.$

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).

[edit] 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. Thus the function r(x) has poles in $x j = c o s ((2 i - 1)π / 2 n)$ 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 due to the fact that p(x) also has a root in zero). We conclude that $I_{-1}^1r=0=I_{-\infty}^{+\infty}r$ since q(x) has only 5 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).