Pade approximants: a case study (iii)

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

The study (iii)

The limit n\to\infty for the Padé approximants P^n_n(z) and P^n_{n+1}(z) exist for all z\in\mathbb{C}\setminus[1,\infty). This is much stronger than the Taylor series, which only converges inside the unit disk and diverges outside.

On the line [1,\infty) the limit does not exist for P^n_{n}(z), P^n_{n+1}(z), where the original function f has a logarithmic branch cut. This fact is “captured” by the Padé rational functions by a funny effect: P^n_{n}(z) and P^n_{n+1}(z) have poles for z>1 which become dense in z\in(1,\infty). E.g. P^{10}_{10}(z):


Although divergent, in some instances the Padé’s still exhibit a rich structure. E.g. P^n_n(2) has two limit points at \pm\frac{\pi}{2}:

pade_2The difference is the same as the discontinuity of f 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.





About goobypl5

pizza baker, autodidact, particle physicist
This entry was posted in Math, Mathematica and tagged , , , . Bookmark the permalink.

One Response to Pade approximants: a case study (iii)

  1. B. Doyle says:

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

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