text: The velocity vector  of the fluid may be written in terms of the stream function  as
  The stream function in a Stokes flow problem,  satisfies the biharmonic equation. By regarding the -plane as the complex plane, the problem may be dealt with using methods of complex analysis. In this approach,  is either the real or imaginary part of
  Here , where  is the imaginary unit, , and  are holomorphic functions outside of the disk. We will take the real part without loss of generality.
  Now the function , defined by  is introduced.  can be written as , or  (using the Wirtinger derivatives).
  This is calculated to be equal to
  Without loss of generality, the disk may be assumed to be the unit disk, consisting of all complex numbers z of absolute value smaller or equal to 1.
  The boundary conditions are:
  whenever ,
  and by representing the functions  as Laurent series:
  the first condition implies  for all .
  Using the polar form of  results in .
  After deriving the series form of u, substituting this into it along with , and changing some indices, the second boundary condition translates to
  Since the complex trigonometric functions  compose a linearly independent set, it follows that all coefficients in the series are zero.
  Examining these conditions for every  after taking into account the condition at infinity shows that  and  are necessarily of the form
  where  is an imaginary number (opposite to its own complex conjugate), and  and  are complex numbers. Substituting this into  gives the result that  globally, compelling both  and  to be zero. Therefore, there can be no motion – the only solution is that the cylinder is at rest relative to all points of the fluid.