The ground state binding energies of axial hydrogenic impurities in a coaxial cylindrical quantum well wire are reported as a function of the barrier height and the radius of wire in the presence of a uniform magnetic field applied parallel to the wire axis. The quantum well wire (QWW) is assumed to be an infinitely long cylinder of GaAs material surrounded by AlxGa1-xAs (for finite case and vacuum for infinite case). Binding energy calculations were performed with the use of a variational procedure in the effective mass approximation. We observed that the binding energy is sensitive to well radius only for both larger R values and small magnetic fields. We also compared the infinite and finite case binding energies and showed that increasing the Al concentration in the finite barrier case, binding energies are increased as expected. Our results are in good agreement and complementary with the previous theoretical works. (C) 2006 Elsevier Ltd. All rights reserved.