Paper 3, Section I, A

(a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.

(b) Find the most general harmonic polynomial of the form

$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$

where $a, b, c, d, x$ and $y$ are real.

(c) Write down a complex analytic function of $z=x+i y$ of which $u(x, y)$ is the real part.

*Typos? Please submit corrections to this page on GitHub.*