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 using one-sided limits as:
:$I_sr = \begin +1, & \text \displaystyle\lim_r(x)=-\infty \;\land\; \lim_r(x)=+\infty, \\ -1, & \text \displaystyle\lim_r(x)=+\infty \;\land\; \lim_r(x)=-\infty, \\ 0, & \text \end$

* 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/math> 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:
:$r(x)=\frac=\frac.$
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 $I_r=I_r=1$ and $I_r=I_r=-1$. For the pole in zero, we have $I_r=0$ since the left and right limits are equal (which is because ''p''(''x'') also has a root in zero).
We conclude that $I_^1r=0=I_^{+\infty}r$ since ''q''(''x'') has only five roots, all in minus;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

Mathematical analysis