We are now going to explore the analytic structure of the Padé approximants for our example [1], [2].

## The study (iii)

The limit for the Padé approximants and exist for all . This is much stronger than the Taylor series, which only converges inside the unit disk and diverges outside.

On the line the limit does not exist for , , where the original function has a logarithmic branch cut. This fact is “captured” by the Padé rational functions by a funny effect: and have poles for which become dense in . E.g. :

Although divergent, in some instances the Padé’s still exhibit a rich structure. E.g. has two limit points at :

The difference is the same as the discontinuity of across the branch cut.

The esoteric conclusion: the Padé’s are much more efficient than the Taylor approximations and “somehow know” about the analytic properties of the original function.

At first sight I was thinking this post was about the ubiquitous Gamma function, but it turned out to be far more interesting!