König's theorem (complex analysis)
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.
Statement
Given a meromorphic function defined on :
Suppose it only has one simple pole in this disk. If such that , then
In particular, we have
Intuition
Near x=r we expect the function to be dominated by the pole:
Matching the coefficients we see that .
References
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