Cauchy index

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 [−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

References

1. "The Cauchy Index".