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, :
:$f(x) = \sum_^\infty c_nx^n, \qquad c_0\neq 0.$
which only has one simple pole $x=r$ in this disk. Then
:$\frac = r + o(\sigma^),$
where $0<\sigma<1$ such that $, r, <\sigma R$. In particular, we have
:$\lim_ \frac = r.$

Intuition

Recall that
:\frac=-\frac\,\frac=-\frac\sum_^\left frac\rightn,
which has coefficient ratio equal to $\frac=r.$

Around its simple pole, a function $f(x) = \sum_^\infty c_nx^n$ 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.
:$f(x)\approx\frac,$
so that $\frac\approx r$.

References

{{DEFAULTSORT:Konig's theorem (complex analysis)
Theorems in complex analysis