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

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