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A two-and-a-half dimensional full-wave forward solver to compute the three-dimensional (3D) electromagnetic field scattered by an infinitely long inhomogeneous (lossy) dielectric cylinder with arbitrary cross-sectional shape under a given 3D time-harmonic illumination is presented. The relevant set of linear equations, obtained after performing a spatial Fourier transform of the fields along the axial direction and applying a method of moments discretization to the two-dimensional contrast source volume integral equation, is solved iteratively with a stabilized biconjugate gradient fast Fourier transform method. In this way, objects with cross-sectional dimensions of several to many wavelengths can be handled in a very fast way. Furthermore, a vectorial 3D Gaussian beam illumination, usually employed in active millimeter-wave (mm-wave) imaging systems, is implemented using a complex source Gaussian beam formulation. The validity of the method is proved by comparison with analytic results and with the results from a full-wave 3D solver. Finally, the method is applied to a millimeter-wave imaging example, showing the scattering from an object hidden under clothing on a human body.