In complex analysis and numerical analysis, Kőnig's theorem, 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 {\displaystyle |x|<R} :
f ( x ) = ∑ n = 0 ∞ c n x n , c 0 ≠ 0. {\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}x^{n},\qquad c_{0}\neq 0.}
which only has one simple pole x = r {\displaystyle x=r} in this disk. Then
c n c n + 1 = r + o ( σ n + 1 ) , {\displaystyle {\frac {c_{n}}{c_{n+1}}}=r+o(\sigma ^{n+1}),}
where 0 < σ < 1 {\displaystyle 0<\sigma <1} such that | r | < σ R {\displaystyle |r|<\sigma R} . In particular, we have
lim n → ∞ c n c n + 1 = r . {\displaystyle \lim _{n\rightarrow \infty }{\frac {c_{n}}{c_{n+1}}}=r.}
Intuition Recall that
C x − r = − C r 1 1 − x / r = − C r ∑ n = 0 ∞ [ x r ] n , {\displaystyle {\frac {C}{x-r}}=-{\frac {C}{r}}\,{\frac {1}{1-x/r}}=-{\frac {C}{r}}\sum _{n=0}^{\infty }\left[{\frac {x}{r}}\right]^{n},}
which has coefficient ratio equal to 1 / r n 1 / r n + 1 = r . {\displaystyle {\frac {1/r^{n}}{1/r^{n+1}}}=r.}
Around its simple pole, a function f ( x ) = ∑ n = 0 ∞ c n x n {\displaystyle f(x)=\sum _{n=0}^{\infty }c_{n}x^{n}} will vary akin to the geometric series and this will also be manifest in the coefficients of f {\displaystyle f} . In other words, near x=r we expect the function to be dominated by the pole, i.e.
f ( x ) ≈ C x − r , {\displaystyle f(x)\approx {\frac {C}{x-r}},}
so that c n c n + 1 ≈ r {\displaystyle {\frac {c_{n}}{c_{n+1}}}\approx r} .
References
