The problem of stress analysis of a circular cylinder is considered under conditions which give rise to stress singularities. Such stresses can be produced by circular boundary discontinuities or when boundaries are subject to mixed displacement and force conditions. General solutions are obtained using transformation of the cylindrical coordinates and these solutions converge in the singular region. The results predict exactly the singular behavior and the analysis generates solutions which can be used over a finite region of the cylinder. A numerical example is given showing the application of this analysis to a circular cylinder completely clamped at one end and loaded axially at the other. In this example the stresses and displacements from the present solution are matched with the corresponding quantities obtained from a finite-element analysis. The matching is performed at a finite distance from the stress singularity. Good agreement is obtained between the two solutions.