extending a harmonic function and finding a non-trivial example

Question

Suppose $u$ is a harmonic function on $\Delta(0,1)$ which extends continuously to $\{x^2+y^2\leq 1: x>0\}$ and is constant on $\{x^2+y^2=1| x>0\}$. Show that $u$ extends to a function whch is harmonic in a neighbourhood of $(1,0)$. What is the largest open set to which $u$ extends as a harmonic function? Can you give a non trivial example of such a function?

Answer 1

Using the Riemann Mapping Theorem we take the unit disc to the upper half plane and map the right boundary of the disc to the interval $(-1,1)$ on the real axis. We can apply the Schwarz Reflection Principle and extend the harmonic function harmonically to the entire strip $\{z=x+iy | |x|<1\}$. By defining $u(\bar{z})=u(z)$ we can in fact extend it to the entire complex plane except the part of the real axis where $|x|>1$.

A possible non-trivial example can be $u(z)=\Re(\sqrt{z^2-1})$ where we choose the branch cuts accordingly.