In complex analysis and numerical analysis, Kőnig'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 :
which only has one simple pole in this disk. Then
where such that . In particular, we have
Intuition Recall that
which has coefficient ratio equal to
Around its simple pole, a function will vary akin to the geometric series and this will also be manifest in the coefficients of .
In other words, near x=r we expect the function to be dominated by the pole, i.e.
so that .
References