## 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 $\pm\frac{\pi}{2}$:

The 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.