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 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 $I\_s$ 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. Thus the function "r"("x") has poles in $x\_j=cos((2i-1)pi/2n)$ 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 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).

External links

* [http://deslab.mit.edu/DesignLab/itango/multi/sld008.htm The Cauchy Index]