Determine whether the following statements are true or false. Justify your answer.
Let \(G = \{z \in \mathbb{C} : |z| < 2 \text{ and } |z-1| > 1\}\). Then the set of polynomials is dense in the space \(H(G)\) of analytic functions on \(G\).
Let \(u_1, u_2\) be harmonic functions on \(\mathbb{D}\). Then \(u := \max\{u_1, u_2\}\) is also harmonic on \(\mathbb{D}\).
Solution
True. By Runge’s theorem, polynomials are dense in \(H(G)\) exactly when the complement of \(G\) in the Riemann sphere is connected. Here \[ G=\{|z|<2\}\cap\{|z-1|>1\}. \] Its complement is the union of \[ \{|z|\ge2\}\cup\{\infty\} \] and \[ \{|z-1|\le1\}. \] These two closed sets meet at \(z=2\), so the complement is connected. Hence polynomials are dense in \(H(G)\).
False. Take \[ u_1(x+iy)=x, \qquad u_2(x+iy)=-x. \] Both are harmonic, but \[ \max\{u_1,u_2\}=|x| \] is not even differentiable on the imaginary axis and therefore is not harmonic.