In complex analysis and numerical analysis, König's theorem,[1] named after the Hungarian mathematician Gyula Kőnig, gives a way to estimate simple poles or simple roots of a function. In particular, it has numerous applications in root finding algorithms like Newton's method and its generalization Householder's method.
Statement
Given a meromorphic function defined on
:
![{\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}x^{n},\qquad c_{0}\neq 0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which only has one simple pole
in this disk. Then
![{\displaystyle {\frac {c_{n}}{c_{n+1}}}=r+o(\sigma ^{n+1}),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
such that
. In particular, we have
![{\displaystyle \lim _{n\rightarrow \infty }{\frac {c_{n}}{c_{n+1}}}=r.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Intuition
Recall that
![{\displaystyle {\frac {C}{x-r}}=-{\frac {C}{r}}\,{\frac {1}{1-x/r}}=-{\frac {C}{r}}\sum _{n=0}^{\infty }\left[{\frac {x}{r}}\right]^{n},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which has coefficient ratio equal to ![{\displaystyle {\frac {1/r^{n}}{1/r^{n+1}}}=r.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Around its simple pole, a function
will vary akin to the geometric series and this will also be manifest in the coefficients of
.
In other words, near x=r we expect the function to be dominated by the pole, i.e.
![{\displaystyle f(x)\approx {\frac {C}{x-r}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
so that
.
References
- ^ Householder, Alston Scott (1970). The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. p. 115. LCCN 79-103908.