Question 4

Let $u(x, y) = - x^3+3xy^2+ 3x^2y- y^3+8x^2-8y^2- 6x- 6y$, defined for any real $(x, y)$.

• Prove that $u$ is harmonic.
• Find the harmonic conjugate $v$ of $u$.
• If $f(z) = u(x, y)+ iv(x, y)$, where $z = x + iy$, explain why $f(z)$ is analytic and write formulas for $f(z)$, $f'(z)$ and $\int f(z) dz$ as polynomials in the variable $z$.
  Given that $f(0) = 0$, explain why $f(z)$ is then unique.
• Also, with $f(z)$ such that $f(0) = 0$, determine the integral:
  $$\mathcal{J} = \frac{1}{2\pi i}\int_\Gamma \frac{f'(z) dz}{f(z)}.$$
  Here the contour $\Gamma$ is the circle $\vert z + 1 - i\vert = 4$, traced once around counter-clockwise.
  (Hint: first find the roots of $f(z)$).
• What can you say about the value of the integral:
  $$\mathcal{K} = \frac{1}{2\pi i} \int_{\Gamma} \frac{zf'(z)}{f(z)} dz.$$