König's theorem (complex analysis)

In complex analysis and numerical analysis, König's theorem, 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 [math]\displaystyle{ |x|\lt R }[/math]:

[math]\displaystyle{ f(x) = \sum_{n=0}^\infty c_nx^n, \qquad c_0\neq 0. }[/math]

which only has one simple pole [math]\displaystyle{ x=r }[/math] in this disk. Then

[math]\displaystyle{ \frac{c_n}{c_{n+1}} = r + o(\sigma^{n+1}), }[/math]

where [math]\displaystyle{ 0\lt \sigma\lt 1 }[/math] such that [math]\displaystyle{ |r|\lt \sigma R }[/math]. In particular, we have

[math]\displaystyle{ \lim_{n\rightarrow \infty} \frac{c_n}{c_{n+1}} = r. }[/math]

Intuition

Recall that

[math]\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, }[/math]

which has coefficient ratio equal to [math]\displaystyle{ \frac{1/r^n}{1/r^{n+1}}=r. }[/math]

Around its simple pole, a function [math]\displaystyle{ f(x) = \sum_{n=0}^\infty c_nx^n }[/math] will vary akin to the geometric series and this will also be manifest in the coefficients of [math]\displaystyle{ f }[/math].

In other words, near x=r we expect the function to be dominated by the pole, i.e.

[math]\displaystyle{ f(x)\approx\frac{C}{x-r}, }[/math]

so that [math]\displaystyle{ \frac{c_n}{c_{n+1}}\approx r }[/math].