This paper is the first of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for a homogeneous finite cylinder that is illuminated at an arbitrary location. Three solutions were derived for an incident delta -light source in the steady-state, frequency, and time domains, respectively, applying different integral transformations. The performance of these solutions was compared with respect to accuracy and speed. Excellent agreement between the solutions, of which some are very fast (< 10ms), was found. Six of the nine solutions were extended to a circular flat beam which is incident onto the top side. Furthermore, the validity of the solutions was tested against Monte Carlo simulations.