A Numerical Test of Pade Approximation for Some Functions with singularity

TL;DR

This paper investigates the effectiveness of Pade approximation in representing functions with various singularities, revealing how poles and zeros distribute around branch cuts and natural boundaries, and assessing numerical accuracy through residue calculus.

Contribution

It provides new insights into the behavior of Pade approximation for functions with branch cuts and natural boundaries, including pole-zero distribution patterns and methods to verify approximation accuracy.

Findings

Poles and zeros align along branch cuts for functions with branch cuts.

Poles distribute around natural boundaries in lacunary and random power series.

Residue calculus can verify numerical accuracy and quasianalyticity.

Abstract

The aim of this study is to examine some numerical tests of Pade approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be specified by the poles of the Pade approximation. However, it have not necessarily been clear how the Pade approximation work for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Pade approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously has a natural boundary on the unit circle. On the other hand, spurious poles and zeros (Froissart doublets) due to numerical errors and/or external noise also appear around the unit circle in the Pade approximation. It is also shown that the residue calculus for the Pade approximated functions can be used to confirm the numerical accuracy of the Pade approximation and quasianalyticity of the random power series.