Computable topology

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 $\left|x\right|<R$:

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n,c_0\ne 0.$

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

$\frac{c_n}{c_{n+1}}=r+o({\sigma }^{n+1}).$

In particular, we have

$\underset{n\to \mathrm{\infty }}{\lim }\frac{c_n}{c_{n+1}}=r.$

Intuition

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

$f(x)\approx \frac{C}{x-r}=-\frac{C}{r}\frac{1}{1-x/r}=-\frac{C}{r}\sum _{n=0}^{\mathrm{\infty }}{\left[\frac{x}{r}\right]}^n.$

Matching the coefficients we see that $\frac{c_n}{c_{n+1}}\approx r$.

References

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