Cauchy Index - Definition

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 is direct (when neither a nor b are poles of r(x)): it is the sum of the Cauchy indices of r for each s located in the interval. We usually denote it by .
  • 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 for a and b going to infinity).

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