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).

Read more about this topic:  Cauchy Index

Famous quotes containing the word definition:

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)