König's theorem (complex analysis) - Alchetron, the free social encyclopedia

In complex analysis and numerical analysis, König'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 :

f ( x ) = ∑ n = 0 ∞ c n x n , c 0 ≠ 0.

Suppose it only has one simple pole x = r in this disk. If 0 < σ < 1 such that | r | < σ R , then

c n c n + 1 = r + o ( σ n + 1 ) .

In particular, we have

lim n → ∞ c n c n + 1 = r .

Intuition

Near x=r we expect the function to be dominated by the pole:

f ( x ) ≈ C x − r = − C r 1 1 − x / r = − C r ∑ n = 0 ∞ [ x r ] n .

Matching the coefficients we see that c n c n + 1 ≈ r .

References

König's theorem (complex analysis) Wikipedia

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Contents

Statement

Intuition

References