Table of Contents
International Journal of Computational Mathematics
Volume 2014 (2014), Article ID 587430, 17 pages
http://dx.doi.org/10.1155/2014/587430
Research Article

A Numerical Test of Padé Approximation for Some Functions with Singularity

1Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata 950-2002, Japan
2Department of Physics, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu 525, Japan

Received 17 July 2014; Accepted 14 October 2014; Published 20 November 2014

Academic Editor: Don Hong

Copyright © 2014 Hiroaki S. Yamada and Kensuke S. Ikeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this study is to examine some numerical tests of Padé 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 characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works 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 Padé 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 have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.