König's theorem (complex analysis)

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 $|x|<R$:

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

Suppose it only has one simple pole $x=r$ in this disk. If $0<\sigma<1$ such that $|r|<\sigma R$, then

$\frac{c_n}{c_{n+1}} = r + o(\sigma^{n+1}).$

In particular, we have

$\lim_{n\rightarrow \infty} \frac{c_n}{c_{n+1}} = r.$

Intuition

Near x=r we expect the function to be dominated by the pole:

$f(x)\approx\frac{C}{x-r}=-\frac{C}{r}\,\frac{1}{1-x/r}=-\frac{C}{r}\sum_{n=0}^{\infty}\left[\frac{x}{r}\right]^n.$

Matching the coefficients we see that $\frac{c_n}{c_{n+1}}\approx r$.

References

1. Householder, Alston Scott (1970). The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. p. 115. LCCN 79-103908.