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 $$|x|<R$$:
 * $$f(x) = \sum_{n=0}^\infty c_nx^n, \qquad c_0\neq 0.$$

which only has one simple pole $$x=r$$ in this disk. Then
 * $$\frac{c_n}{c_{n+1}} = r + o(\sigma^{n+1}),$$

where $$0<\sigma<1$$ such that $$|r|<\sigma R$$. In particular, we have
 * $$\lim_{n\rightarrow \infty} \frac{c_n}{c_{n+1}} = r.$$

Intuition
Recall that
 * $$\frac{C}{x-r}=-\frac{C}{r}\,\frac{1}{1-x/r}=-\frac{C}{r}\sum_{n=0}^{\infty}\left[\frac{x}{r}\right]^n,$$

which has coefficient ratio equal to $$\frac{1/r^n}{1/r^{n+1}}=r.$$

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

In other words, near x=r we expect the function to be dominated by the pole, i.e.
 * $$f(x)\approx\frac{C}{x-r},$$

so that $$\frac{c_n}{c_{n+1}}\approx r$$.