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Previous: Schwartz's lemma
Residues can show up in defining fixed points for rational functions. Suppose
that the function r,
r(z) |
= |
![$\displaystyle \frac{P(z)}{Q(z)},$](img281.gif) |
(173) |
is a quotient of two polynomials. If Q has lesser degree than P, division may be used to write
r(z) |
= |
![$\displaystyle g(z) + \frac{p(z)}{q(z)}$](img282.gif) |
(174) |
wherein p and q have the desired property.
Consider a ``resolvent'' R(z) defined by
R(z) |
= |
![$\displaystyle \frac{1}{r(z)-z}$](img283.gif) |
(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) |
= |
![$\displaystyle \frac{1}{g(z)+\frac{p(z)}{q(z)}-z}$](img284.gif) |
(176) |
|
= |
![$\displaystyle \frac{q(z)}{g(z)q(z)+p(z)-zq(z)}.$](img285.gif) |
(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) |
= |
![$\displaystyle \sum \frac{a_j}{z - z_j}$](img286.gif) |
(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) |
= |
![$\displaystyle \frac{1}{R(z)}+z$](img287.gif) |
(179) |
r'(zf) |
= |
![$\displaystyle \lim_{z\rightarrow z_f}
\frac{1}{z-z_f}\left(\frac{1}{R(z)}+z\right)$](img288.gif) |
(180) |
|
= |
![$\displaystyle 1 + \frac{1}{a_j}.$](img289.gif) |
(181) |
So stability depends on negative (but not too negative) aj's.
Next: representation of a function
Up: Contour Integrals
Previous: Schwartz's lemma
Microcomputadoras
2001-04-05