  We develop a method to analyze the strong coupling limit of the Bethe ansatz
equations supposed to give the spectrum of anomalous dimensions of the planar
${\cal N}=4$ gauge theory. This method is particularly adapted for the three
rank-one sectors, $su(2)$, $su(1|1)$ and $sl(2)$. We use the elliptic
parametrization of the Bethe ansatz variables, which degenerates to a
hyperbolic one in the strong coupling limit. We analyze the equations for the
highest excited states in the su(2) and $su(1|1)$ sectors and for the state
corresponding to the twist-two operator in the $sl(2)$ sector, both without and
with the dressing kernel. In some cases we were able to give analytic
expressions for the leading order magnon densities. Our method reproduces all
existing analytical and numerical results for these states at the leading
order.
