  The reconfigurable intelligent surface (RIS), which consists of a large
number of passive and low-cost reflecting elements, has been recognized as a
revolutionary technology to enhance the performance of future wireless
networks. This paper considers an RIS assisted multicast transmission, where a
base station (BS) with multiple-antenna multicasts common message to multiple
single-antenna mobile users (MUs) under the assistance of an RIS. An equivalent
channel model for the considered multicast transmission is analyzed, and then
an optimization problem for the corresponding channel capacity is formulated to
obtain the optimal covariance matrix and phase shifts. In order to solve the
above non-convex and non-differentiable problem, this paper first exploits the
gradient descent method and alternating optimization, to approach the locally
optimal solution for any number of MUs. Then, this paper considers a special
case, which can obtain the global optimal solution, and shows the sufficient
and necessary condition for this special case. Finally, the order growth of the
maximal capacity is obtained when the numbers of the reflecting elements, the
BS antennas, and the MUs go to infinity.
