As mentioned above, the stream function and velocity potential are expressed as multipole expansions. The coefficients of the multipole expansion, p_2m and q_2m, are found by applying the appropriate boundary condition at the cylinder surface. This leads to Equation ( 47 ).

This may be re-arranged and expressed in matrix form:

Where the vector x contains the p_2m or q_2m terms, the matrix A contains the f_2m terms and the vector b contains the Y_c or Y_s terms.

The terms in Equation ( 47 ) are evaluated as follows:

The mapped points y, z are obtained by applying the mapping equation at angle q (Equation( 49 )).

Since the integral in Equation ( 51 ) converges slowly it is evaluated by an alternative method. The method used follows the work of Sutherland and is known as the method of Laguerre-Gauss quadrature. It may be shown that the integral can be evaluated as in Equation ( 52 ):

Where the weighting functions, w_i, and the abscissa, s_i, may be found in standard texts. Finally the f_2m terms are calculated for each multipole at each angle according to Equation ( 53 ):

with y_2m:

where a₀, a₁, ... a_N are the conformal mapping coefficients.