can be found by setting

Here is an analytic function of the complex variable . Splitting the function into real and imaginary parts we obtain

and both and satisfy Laplace's equation.

__Example 5. .2__Consider solutions to Laplace's equation with

In this case we may try

Expanding the square and expressing in terms of real and imaginary parts we obtain

Thus, we have

and

Using the real part for the potential, , the solution to Laplace's equation is

It is straightforward to show that both Laplace's equation and the boundary conditions are satisfied.