  We study the asymptotic development at infinity of an integral operator. We
use this development to give sufficient conditions in order to upper bound the
number of critical periodic orbits that bifurcate from the outer boundary of
the period function of planar potential centers. We apply the main results to
two different families: the power-like potential family $\ddot x=x^p-x^q$,
$p,q\in\mathbb{R}$, $p>q$; and the family of dehomogenized Loud's centers.
