Consider the function \(u(x, y)=x^{3}-3 x y^{2}\).

a. Show that \(u(x, y)\) is harmonic; that is, \(abla^{2} u=0\).

b. Find its harmonic conjugate, \(v(x, y)\).

c. Find a differentiable function, \(f(z)\), for which \(u(x, y)\) is the real part.

d. Determine \(f^{\prime}(z)\) for the function in part

c. [Use \(f^{\prime}(z)=\frac{\partial u}{\partial x}+i \frac{\partial v}{\partial x}\) and rewrite your answer as a function of \(z\).]