How to choose contour for rational approximation Let $f$ be an analytic function on $\Omega \subset \mathbb{C}$. The Hermite formula for interpolation at the points $a_k$, $k=1,\ldots,n$, using a rational function $r_n$ with poles at $b_k$, $k=1,\ldots,n$,
$$
        f(z) - r_n(z) 
        =
        \frac{1}{2\pi i} \oint_{\partial \Omega} \frac{\ell_n(z)}{\ell_n(w)} \, \frac{f(w)}{w-z} \, dw
, \qquad
\ell_n(z) = \frac{\prod_{k = 1}^n (z-a_k)}{\prod_{k=1}^n (z-b_k)}
,
$$
allows to estimate the interpolation error on a compact set $A\subset\Omega$ by 
$$
        \|f - r_n\|_{\infty,A}
        \leq
        \|f\|_{\infty,A} \, \frac{\sup_{z \in A} |\ell_n(z)|}{\inf_{w \in \partial\Omega} |\ell_n(w)|}
.
$$
We can then use logarithmic potential theory to find $a_k$ and $b_k$ which asymptotically minimise this expression, see e.g. chapter VIII in Saff & Totik "Logarithmic Potentials with External Fields". Thus, there is a complete theory on how to do asymptotically optimal rational interpolation except that I have to a-priori choose the domain of analyticity $\Omega$ or equivalently the contour $\partial \Omega$. Are there any guidelines/theory on how to do this?
 A: If the points $\{a_{k}\}_{k=1}^{n}\subset A$ and $\{b_{k}\}_{k=1}^{n}\subset\mathbb C\setminus\Omega$ are chosen as rational Fekete points (see p.396 of the reference mentioned by the OP), then it can be proved (see p.397) that
$$\limsup_{n\infty}\left(\frac{\sup_{z\in A}|l(z)|}{\inf_{z\in\mathbb C\setminus\Omega}|l(z)|}\right)^{1/n}\leq e^{-1/\text{cap}(A,\mathbb C\setminus\Omega)},$$
where $\text{cap}(A,\mathbb C\setminus\Omega)$ denotes the capacity of the condenser made of the two plates $A$ and $\mathbb C\setminus\Omega$, which is defined as the inverse of
$$\inf_{\mu_{1},\mu_{2}}\iint\log\frac{1}{|z-t|}d(\mu_{1}-\mu_{2})(z)d(\mu_{1}-\mu_{2})(t),$$
where $\mu_{1}$ and $\mu_{2}$ are probability measures supported on $A$ and 
$\mathbb C\setminus\Omega$ respectively. Note that if one of the sets becomes smaller, the infimum increases and the capacity decreases.
In order to optimize the rate of approximation, one thus has to minimize the capacity, or equivalently to choose $\Omega$ as large as possible. Hence the best rate of approximation is achieved when choosing $\Omega$ as the (maximal) domain of analyticity of the function $f$ and the contour as the boundary of this domain.

To answer the question in the comment about rational approximation to the function $1/(1+\exp(x))$ on the interval $[-1,1]$, here are the plots of the error functions of best (minimax) rational approximants of degrees 3 and 5, as computed by the Remez command of the Chebfun package.

