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.
Contents
Statement
Given a meromorphic function defined on
Suppose it only has one simple pole
In particular, we have
Intuition
Near x=r we expect the function to be dominated by the pole:
Matching the coefficients we see that