Cauchy index explained

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]

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 , & if \displaystyle\lim _{x\uparrow} r(x) =-inf ty \land \lim _{x\downarrow} r(x) =+inf ty, \\ -1 , & if \displaystyle\lim _{x\uparrow} r(x) =+inf ty \land \lim _{x\downarrow} r(x) =-inf ty, \\ 0 , & 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 ^{br} _{a} .

• We can then generalize to intervals of type

[ -inf ty, +inf ty] 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) = (4 x ^{3} -3 x)/(= 16 x ^{5} -20 x ^{3} +5 x) (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 ^{+inf ty} _{-inf ty} 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

Notes and References

1. Web site: The Cauchy Index . 2024-01-20 . deslab.mit.edu.