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 :
which only has one simple pole in this disk. Then
where such that. In particular, we haveIntuition
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.
