Let the function f (z) = u(r, ) + iv(r, ) be analytic in a domain D

Let the function f (z) = u(r, θ) + iv(r, θ) be analytic in a domain D that does not include the origin. Using the Cauchy-Riemann equations in polar coordinates (Sec. 23) and assuming continuity of partial derivatives, show that throughout D the function u(r, θ) satisfies the partial differential equation
r2urr (r, θ) + rur (r, θ) + uθθ (r, θ) = 0,
which is the polar form of Laplace's equation. Show that the same is true of the function v(r, θ).