Verify that the function u(r, ) = ln r is harmonic in the domain r > 0, 0 < < 2 by showing that

Verify that the function u(r, θ) = ln r is harmonic in the domain r > 0, 0 < θ < 2π by showing that it satisfies the polar form of Laplace's equation, obtained in Exercise 5. Then use the technique in Example 5, Sec. 26, but involving the Cauchy-Riemann equations in polar form (Sec. 23), to derive the harmonic conjugate v(r, θ) = θ. (Compare with Exercise 6, Sec. 25.)