A simple method for finding the particular integral for  in two dimensions was devised by Isao Imai and Ernst Lamla. In two dimensions, the problem can be handled using complex analysis by introducing the complex potential  formally regarded as the function of  and its conjugate ; here  is the stream function, defined such that
where  is some reference value for the density. The perturbation series of  is given by
where  is an analytic function since  and , being solutions of the Laplace equation, are harmonic functions. The integral for the first-order problem leads to the Imai–Lamla formula
where  is the homogeneous solution (an analytic function), that can be used to satisfy necessary boundary conditions. The series for the complex velocity potential  is given by
where  and