  This paper is concerned with the analytic behaviors (monotonicity,
isochronicity and the number of critical points) of period function for
potential system $\ddot{x}+g(x)=0$.We give some sufficient criteria to
determine the monotonicity and upper bound to the number of critical periods.
The conclusion is based on the semi-group properties of (Riemann-Liouville)
fractional integral operator of order $\frac{1}{2}$ and Rolle's Theorem. In
polynomial potential settings, bounding the the number of critical periods of
potential center can be reduced to counting the real zeros of a semi-algebraic
system. From which we prove that if nonlinear potential $g$ is odd, the
potential center has at most $\frac{deg(g)-3}{2}$ critical periods. To
illustrate its applicability some known results are proved in more efficient
way, and the critical periods of some hyper-elliptic Hamiltonian systems of
degree five with complex critical points are discussed, it is proved the system
can have exactly two critical periods.
