  In this work we study the analytical properties of a 2+1 dimensional
magnetically charged holographic superconductor in $AdS_4$. We obtain the
critical chemical potential $\mu_c$ analytically, using the Sturm-Liouville
variational approach. Also, the obtained analytic result can be used to back up
the numerical computations in the holographic superconductor in the probe
limit.
