On the Asymptotics of the Rows of the Pad\'e Table of Analytic Functions with Logarithmic Branch Points

A. P. Starovoitov ; N. A. Starovoitova

Matematičeskie zametki, Tome 84 (2008) no. 3, pp. 409-419

Résumé

For the functions $f(z)=\sum_{n=0}^\infty z^{l_n}/a_n$, where $l_n$ and $a_n$ are arithmetic progressions and their Padé approximants $\pi_{n,m}(z;f)$, we establish an asymptotics of the decrease of the difference $f(z)-\pi_{n,m}(z;f)$ for the case in which $z\in D=\{z:|z|1\}$, $m$ is fixed, and $n\to\infty$. In particular, we obtain proximate orders of decrease of best uniform rational approximations to the functions $\ln(1-z)$ and $\operatorname{arctan}z$ in the disk $D_q=\{z:|z|\le q1\}$.

Détail

Mots-clés : Padé approximant, Padé table, Hadamard determinant
Keywords: analytic function, logarithmic branch point, arithmetic progression, Vandermonde determinant.