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> Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$
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My attempt:

I understand that $u$ is real part of $\text{log}(z)$ and $\text{log}(z)$ is not well defined on $\Omega.$ How do I use this fact and identity theorem to show there isn't any holomorphic function on $\Omega$ whose real part is $u \ ?$

Thank you.