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 ${\displaystyle |x|<R}$:

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

which only has one simple pole ${\displaystyle x=r}$ in this disk. Then

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

where ${\displaystyle 0<\sigma <1}$ such that ${\displaystyle |r|<\sigma R}$. In particular, we have

${\displaystyle \lim _{n\rightarrow \infty }{\frac {c_{n}}{c_{n+1}}}=r.}$

Intuition

Recall that

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

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

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

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

${\displaystyle f(x)\approx {\frac {C}{x-r}},}$

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

References

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