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]

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:

${\displaystyle I_{s}r={\begin{cases}+1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=-\infty \;\land \;\lim _{x\downarrow s}r(x)=+\infty ,\\-1,&{\text{if }}\displaystyle \lim _{x\uparrow s}r(x)=+\infty \;\land \;\lim _{x\downarrow s}r(x)=-\infty ,\\0,&{\text{otherwise.}}\end{cases}}}$

• 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 ${\displaystyle I_{s}}$ of r for each s located in the interval. We usually denote it by ${\displaystyle I_{a}^{b}r}$.
• We can then generalize to intervals of type ${\displaystyle [-\infty ,+\infty ]}$ 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:

${\displaystyle r(x)={\frac {4x^{3}-3x}{16x^{5}-20x^{3}+5x}}={\frac {p(x)}{q(x)}}.}$

We recognize in p(x) and q(x) respectively the Chebyshev polynomials of degree 3 and 5. Therefore, r(x) has poles ${\displaystyle x_{1}=0.9511}$, ${\displaystyle x_{2}=0.5878}$, ${\displaystyle x_{3}=0}$, ${\displaystyle x_{4}=-0.5878}$ and ${\displaystyle x_{5}=-0.9511}$, i.e. ${\displaystyle x_{j}=\cos((2i-1)\pi /2n)}$ for ${\displaystyle j=1,...,5}$. We can see on the picture that ${\displaystyle I_{x_{1}}r=I_{x_{2}}r=1}$ and ${\displaystyle I_{x_{4}}r=I_{x_{5}}r=-1}$. For the pole in zero, we have ${\displaystyle I_{x_{3}}r=0}$ since the left and right limits are equal (which is because p(x) also has a root in zero). We conclude that ${\displaystyle I_{-1}^{1}r=0=I_{-\infty }^{+\infty }r}$ 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

External links

• The Cauchy Index