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.[1]

## Contents

* 1 Definition
* 2 Examples
* 3 References
* 4 External links

## 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:

Isr={+1,if limx↑sr(x)=−∞∧limx↓sr(x)=+∞,−1,if limx↑sr(x)=+∞∧limx↓sr(x)=−∞,0,otherwise.

* 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 Is of r for each s located in the interval. We usually denote it by Iabr.
* We can then generalize to intervals of type [−∞,+∞] 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)=4x3−3x16x5−20x3+5x=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 x1=0.9511, x2=0.5878, x3=0, x4=−0.5878 and x5=−0.9511, i.e. xj=cos⁡((2i−1)π/2n) for j=1,...,5. We can see on the picture that Ix1r=Ix2r=1 and Ix4r=Ix5r=−1. For the pole in zero, we have Ix3r=0 since the left and right limits are equal (which is because p(x) also has a root in zero). We conclude that I−11r=0=I−∞+∞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).

## References

1. ↑ "The Cauchy Index". http://deslab.mit.edu/DesignLab/itango/multi/tsld008.htm.

## External links

* The Cauchy Index

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