Two different methods are used to find the stream function for a rotating cylinder containing a frictionless fluid. In the first method we use complex variables and give a generalization of the well-known results. In the second method we consider suitable distributions of Dirac σ-function or its derivatives on lines. Also the first method is applied to solve the problem of a rotating cylinder with the fluid external to it. The solutions we obtain are in closed form; not only do we obtain some new solutions for certain cylinders but we also recover some of the known solutions in an easy way.