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## residues and the stability of fixed points

Residues can show up in defining fixed points for rational functions. Suppose that the function r,

 r(z) = (173)

is a quotient of two polynomials. If Q has lesser degree than P, division may be used to write
 r(z) = (174)

wherein p and q have the desired property.

Consider a resolvent'' R(z) defined by

 R(z) = (175)

which will have poles wherever r(z) has fixed points. As previously discussed, r(z)-z can be factored, allowing the writing of
 R(z) = (176) = (177)

This denominator has greater degree than the numerator, so it can be factored and R can be written as a sum of partial fractions (due allowance should be made for repeated factors)

 R(z) = (178)

for fixed points of r, namely zj. The aj's are residues of R, but we are interested in the stability of r(z).
 r(z) = (179) r'(zf) = (180) = (181)

So stability depends on negative (but not too negative) aj's.

Next: representation of a function Up: Contour Integrals Previous: Schwartz's lemma