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.
Given a meromorphic function defined on
: which only has one simple pole
in this disk.
0 < σ < 1
< σ
In particular, we have 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