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 $|x|<R$:

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n,c_0\ne 0.$

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

$\underset{n\to \mathrm{\infty }}{\lim }\frac{c_n}{c_{n+1}}=r.$

Intuition

Recall that

$\frac{C}{x-r}=-\frac{C}{r}\frac{1}{1-x/r}=-\frac{C}{r}\sum _{n=0}^{\mathrm{\infty }}{\left[\frac{x}{r}\right]}^n,$

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

Around its simple pole, a function $f(x)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$ 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 $\frac{c_n}{c_{n+1}}\approx r$.