Analytical solutions are developed for turbulent plumes rising from circular sources of positive buoyancy in a quiescent environment of uniform density. From governing equations written in a form which encompasses both the Boussinesq and non-Boussinesq cases, we derive analytical expressions for all plume variables (radius, velocity, and density deficit) in terms of a single quantity Gamma, called the plume function. For given source conditions, we then show that Gamma (and, subsequently, all plume variables) can be evaluated at any height from two integral functions which are defined for lazy and forced plumes. For a practical use, these integral functions are given in tables. Moreover, exact values and locations of the maximum velocity and the plume neck are determined.