  The weak-field expansion of the charged fermion propagator under a uniform
magnetic field is studied. Starting from Schwinger's proper-time
representation, we express the charged fermion propagator as an infinite series
corresponding to different Landau levels. This infinite series is then
reorganized according to the powers of the external field strength $B$. For
illustration, we apply this expansion to $\gamma\to \nu\bar{\nu}$ and $\nu\to
\nu\gamma$ decays, which involve charged fermions in the internal loop. The
leading and subleading magnetic-field effects to the above processes are
computed.
