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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)
Suppose it only has one simple pole
in this disk. If
such that
, then
![{\displaystyle {\frac {c_{n}}{c_{n+1}}}=r+o(\sigma ^{n+1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87139a739087f67a2cd4571dd3c12a5a23e29ced)
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
Near x=r we expect the function to be dominated by the pole:
![{\displaystyle f(x)\approx {\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/653ace3d4d88d670dfdc3791d59b86ac4b45ffc2)
Matching the coefficients we see that
.
References
- ^ Householder, Alston Scott (1970). The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. p. 115. LCCN 79-103908.