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 :

which only has one simple pole in this disk. Then

where such that . In particular, we have

Intuition

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 .

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References

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