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

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Given a meromorphic function defined on  :

 

which only has one simple pole   in this disk. Then

 

where   such that  . In particular, we have

 

Intuition

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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

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  1. ^ Householder, Alston Scott (1970). The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. p. 115. LCCN 79-103908.