I know that if a real function is harmonic, then its derivative is the real part of an holomorphic function, and if we're on simply connected domain, then the function itself is the real part of an analytic function.
Now, $\varGamma(\overline{z} )$ defined on the whole complex plane without the non positive integers. It is not a simply connected domain so I cant use the sentence (and also, I think $\varGamma(\overline{z})$ is not even holomorphic since $\overline{z} $ is not holomorphic).
So how can I determine?