Substituting into Laplace's equation we obtain

On dividing by and rearranging into functions of and functions of alone, this gives

The form of the solution depends on the sign of the separation constant .

- . The solution is

In general, we need to sum over all values of to obtain the most general solution. - . In this case the solution is

- . The solution is somewhat simpler in this case, with
linear solutions for both and .

__Example 5. .1__Consider an irrotational, incompressible flow in a channel of width
. The situation is shown in Figure 5.1

The boundary conditions are that there is no normal flow through the channel walls. Hence,

where is a specified function of . The boundary conditions in the -direction imply that the solution should be oscillatory in while in the -direction they imply that the variation is exponential in . Therefore, we try a solution of the form, is taken as positive,

This satisfies the boundary condition at and the condition for large positive . Hence,

Thus, the general solution is

and

The constants in the summation are given from the condition on at by inverting the Fourier series as