  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.
