Given an incompressible two-dimensional flow which is also irrotational we have:
Which is in cylindrical coordinates 
We look for a solution with separated variables:
which gives
Given the left part depends only on r and the right parts depends only on , the two parts must be equal to a constant independent from r and . The constant shall be positive.
Therefore,
The solution to the second equation is a linear combination of  and 
In order to have a single-valued velocity (and also a single-valued stream function) m shall be a positive integer.
therefore the most generic solution is given by
The potential is instead given by