  We reexamine the process $\gamma\to e^++ e^-$ in a background magnetic field
comparable to $B_c\equiv m_e^2/e$. This process is known to be non-perturbative
in the magnetic-field strength. However, it can be shown that the {\it moments}
of the above pair production width is proportional to the derivatives of photon
polarization function at the zero energy, which is perturbative in $B$. Hence,
the pair-production width can be easily obtained from the latter by the inverse
Mellin transform. The implications of our approach are discussed.
