  A three-dimensional (3D) Poisson solver with longitudinal periodic and
transverse open boundary conditions can have important applications in beam
physics of particle accelerators. In this paper, we present a fast efficient
method to solve the Poisson equation using a spectral finite-difference method.
This method uses a computational domain that contains the charged particle beam
only and has a computational complexity of $O(N_u(logN_{mode}))$, where $N_u$
is the total number of unknowns and $N_{mode}$ is the maximum number of
longitudinal or azimuthal modes. This saves both the computational time and the
memory usage by using an artificial boundary condition in a large extended
computational domain.
