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):

pade10_C

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.

 

 

 

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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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