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

r(z) |
= | (174) |

wherein

Consider a ``resolvent'' *R*(*z*) defined by

R(z) |
= | (175) |

which will have poles wherever

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(z) |
= | (179) | |

r'(z_{f}) |
= | (180) | |

= | (181) |

So stability depends on negative (but not too negative)