Montessus de Ballore's theorem

Remarks

The idea of the proof: let $Q(z) = \prod_{p \in P} (z - p)$, so that $Qf$ is analytic on $D$. The denominators $q_m$ of $[m/n]_f$ are determined by $n$ linear conditions on the Taylor coefficients of $f$ of high order. The asymptotics of these coefficients are dominated by the poles in $P$, and one shows that $q_m \to Q$ after normalization, at a geometric rate. Then $q_m f - p_m$ is small by construction, and dividing by $q_m$ gives the convergence away from $P$. Example: $f(z) = e^z / (1 - z)$ has exactly one pole in every disc of radius $R > 1$. So the column $[m/1]_f$ converges to $f$ on compact subsets of $\C \setminus \set{1}$, and the pole of $[m/1]_f$ tends to $1$. The hypothesis that the number of poles equals $n$ matters. With fewer poles the approximants may have spurious poles, and with more poles a denominator of degree $n$ cannot capture them all. The theorem was proved by Robert de Montessus de Ballore in 1902.

Going deeper

For a student

This is the basic convergence theorem for columns of the Padé table, where the denominator degree is fixed and the numerator degree grows. It gives a practical way to locate the poles of a function from its Taylor coefficients, as the limits of the poles of the approximants. For diagonal sequences $[n/n]_f$, where both degrees grow, convergence is much harder; uniform convergence can fail, which is the subject of the Baker–Gammel–Wills conjecture.