  We present a theoretical study of a nanowire made of a three-dimensional
topological insulator. The bulk topological insulator is described by a
continuum-model Hamiltonian, and the cylindrical-nanowire geometry is modelled
by a hard-wall boundary condition. We provide the secular equation for the
eigenergies of the systems (both for bulk and surface states) and the
analytical form of the energy eigenfunctions. We describe how the surface
states of the cylinder are modified by finite-size effects. In particular, we
provide a $1/R$ expansion for the energy of the surface states up to second
order. The knowledge of the analytical form for the wavefunctions enables the
computation of matrix elements of any single-particle operators. In particular,
we compute the matrix elements of the optical dipole operator, which describe
optical absorption and emission, treating intra- and inter-band transition on
the same footing. Selection rules for optical transitions require conservation
of linear momentum parallel to the nanowire axis, and a change of $0$ or $\pm
1$ in the total-angular-momentum projection parallel to the nanowire axis. The
magnitude of the optical-transition matrix elements is strongly affected by the
finite radius of the nanowire.
