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 $|x|<R$:

$f(x)=\sum _{n=0}^{\infty }c_{n}x^{n},\qquad c_{0}\neq 0.$

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 $f(x)=\sum _{n=0}^{\infty }c_{n}x^{n}$ 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 ${\frac {c_{n}}{c_{n+1}}}\approx r$.