Let u be a real-valued function defined on the unit disc D. Suppose thatu is twice continuously differentiable and harmonic, that is,
∆u(z, y) = 0
for all (r, y) ∈ D
(a) Prove that there exists a holomorphic function f on the unit disc such that
Re(f) = u
Also show that the imaginary part of f is uniquely defined up to an additive (real) constant. [Hint: From the previous chapter we would have f'(z) = 2∂u/∂z Therefore, let g(z) 2∂u/∂z and prove that g is holomorphic. Why can one find F with F'= g? Prove that Re(F) differs from u by a real constant.