Show that arg(tanh z ) is given by tan - 1 (sin 2 y/ sinh 2 x ). Identify arg(tanh z ) as the imaginary part of a suitable function of a complex variable and hence deduce that, for x > 0, tan - 1 (sin 2 y/ sinh 2 x ) is a solution of Laplace's equation. By means of an appropriate scaling, find an explicit solution of Laplace's equation that takes the value 0 on the lines y = 0 and y = a (for x > 0) and the value A on the line x = 0 (for 0 y a ).