On the open set $U := \{ 1 < |z| < 2\} \subset \mathbb C$ the function $u(x,y) = \log(x^2+y^2)$ is $C^\infty$ and harmonic. Prove or disprove that there is an analytic function $F(z)$ on $U$ such that $\Re \ F(x+iy) = u(x,y)$.
So, assuming that such an $F=u(x,y)+iv(x,y)$ exists, then it satisfies the Cauchy Riemann equation and $v$ is harmonic. We can get that $v(y) = 2 \arctan(y/x) + g(x)$ and $v(x) = -2 \arctan(x/y) + h(y)$ for some functions $g,h$. However, I am not sure where to go about from here.
Let Log be the principle branch of log. If $F$ exists as stated then $\Re F(z)=\frac 1 2 \Re Log(z)$ which implies $F(z)=\frac 1 2 Log(z)$ on $\{z:1<|z|<2\}$. with the part in the negative real axis removed (because if two analytic functions in region have the same real part then they are equal). This implies Log extends to an analytic function in $\{z:1<|z|<2\}$. It is well known that this is not true. In fact limit of $Log (z)$ as $z$ approaches -0.5 vertically up and vertically down are different.