  We present a detailed measurement of charged two-pion correlation functions
in 0%-30% centrality $\sqrt{s_{_{NN}}}=200$ GeV Au$+$Au collisions by the
PHENIX experiment at the Relativistic Heavy Ion Collider. The data are well
described by Bose-Einstein correlation functions stemming from L\'evy-stable
source distributions. Using a fine transverse momentum binning, we extract the
correlation strength parameter $\lambda$, the L\'evy index of stability
$\alpha$ and the L\'evy length scale parameter $R$ as a function of average
transverse mass of the pair $m_T$. We find that the positively and the
negatively charged pion pairs yield consistent results, and their correlation
functions are represented, within uncertainties, by the same L\'evy-stable
source functions. The $\lambda(m_T)$ measurements indicate a decrease of the
strength of the correlations at low $m_T$. The L\'evy length scale parameter
$R(m_T)$ decreases with increasing $m_T$, following a hydrodynamically
predicted type of scaling behavior. The values of the L\'evy index of stability
$\alpha$ are found to be significantly lower than the Gaussian case of
$\alpha=2$, but also significantly larger than the conjectured value that may
characterize the critical point of a second-order quark-hadron phase
transition.
