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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cadc37c9a1be079b6dfb4080078c0e0087e6398b)
which only has one simple pole
in this disk. Then
![{\displaystyle {\frac {c_{n}}{c_{n+1}}}=r+o(\sigma ^{n+1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f1ae7f0fb4728b7ce123586213c47ddb3d2cae)
where
such that
. In particular, we have
![{\displaystyle \lim _{n\rightarrow \infty }{\frac {c_{n}}{c_{n+1}}}=r.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/355944997e1ba7fdb66d69fa2e20c77e49311800)
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/085d5431b33eaa878e5a3dcf5dda8dccd71d4671)
which has coefficient ratio equal to
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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd17e60e9e20fe0db402349363c3000c9d67443)
so that
.
References
- ^ Householder, Alston Scott (1970). The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. p. 115. LCCN 79-103908.