Let v(x, y) = x^3 3x(y 2)y, where x, y R.

(i) Verify that v is a harmonic function.

(ii) Find a holomorphic function f : C C that has v as its imaginary part, i.e., find a real-valued function u(x, y) such that f(z) = f(x + iy) = u(x, y) + iv(x, y) for all x, y R.

(iii) Express the function f from part (ii) as a function of z C (not as a function of x and y).