Exact longitudinal equilibrium distribution of stored electrons in the presence of self-fields

The longitudinal equilibrium distribution of electrons stored in a ring is computed in the general case of a nonlinear accelerating field. From this is derived an equation which gives the exact longitudinal density in the presence of longitudinal fields generated by the circulating particles. A method for solving numerically this equation is given and the exact solution is compared to the lowest-approximation solution when self-fields are proportional to the derivative of the bunch density.