König's theorem (complex analysis) explained

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

|x|<R :f(x) = \sum ^{inf ty} _{n =0} c ^{n,} _{nx} c _{0 ≠} 0 . which only has one simple pole x = r in this disk. Then(c _{n})/(c _{n+1}) = r + o(\sigma ^{n+1} ), where 0 <\sigma< 1 such that |r|<\sigma R . In particular, we have\lim _{n →} (c _{n})/(c _{n+1}) = r.

Intuition

Recall that

(C)/(=- x - r) (C)/(r) (1)/(=- 1- x/r) (C)/(r) \sum ^{inf ty} _{n =0} \left[ (x)/(r) \right] ^{n,} which has coefficient ratio equal to (1 /r ^{n})/(1 /r ^{n+1}) = r.

Around its simple pole, a function

f(x) = \sum ^{inf ty} _{n =0} c ^{n} _{nx} will vary akin to the geometric series and this will also be manifest in the coefficients of f .

In other words, near x=r we expect the function to be dominated by the pole, i.e.

f(x) ≈ (C)/(x - r) , so that (c _{n})/(c _{n+1}) ≈ r .

Notes and References

1. Book: Householder, Alston Scott. 1970. The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill. 79-103908. 115.