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We report, in the present work, a numerical method for investigating guided waves propagation in a homogeneous infinite cylinder composed of elastic material. This method makes use of Legendre polynomials series and harmonic function to express different displacement components which are introduced into the equation of motion. The advantage of this method is the possibility to incorporate the stress-free boundary conditions directly into the equations of motion by assuming position-dependent elastic constants and mass density. The solution of the wave equations can be reduced to an eigenvalue problem. Numerical results are presented and compared with those published earlier in order to validate our polynomial approach. For certain specific modes, dispersion curves and field profiles such as mechanical displacements, normal stresses are presented. The developed software is capable of dealing efficiently and accurately with a variety of homogeneous and inhomogeneous cylinders.