The electrostatic potential satisfies Laplace’s equation ∇²u = 0 in any region free of charges. Also the heat equation u_t = c² ∇²u (Sec. 12.5) reduces to Laplace’s equation if the temperature u is time-independent (“steady-state case”). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < 1 if the boundary values are (sketch them, to see what is going on).

u (1, θ) = θ if -1/2π < θ < 1/2π and 0 otherwise